Math in Focus

Practice the problems of Math in Focus Grade 8 Workbook Answer Key Chapter 7 Lesson 7.3 Understanding the Pythagorean Theorem and Solids to score better marks in the exam.

Math in Focus Grade 7 Course 3 B Chapter 7 Lesson 7.3 Answer Key Understanding the Pythagorean Theorem and Solids

Math in Focus Grade 8 Chapter 7 Lesson 7.3 Guided Practice Answer Key

Solve.

Question 1.
The radius of a cone-shaped hat is 5 centimeters. The slant length of the hat is 8 centimeters. What is the height of the hat? Round your answer to the nearest tenth.

Let the height be x centimeters.

So, the height of the hat is approximately centimeters.
Answer:
The height of the hat is approximately 6 centimeters,

Explanation:

Solve.

Question 2.
Given the diagonal length, XY, of the end wall of the King’s Chamber of the Great Pyramid and the length, XZ, of the chamber, find the length, YZ, of the central diagonal. Find both the exact value and the approximate value to the nearest tenth.

+ = YZ² Use the Pythagorean Theorem.
+ = YZ² Substitute values for \(\overline{X Y}\) and \(\overline{X Z}\).
+ = YZ² Multiply.
= Add.
= YZ Find the positive square root.
YZ ≈ m Round to the nearest tenth.
So, the length of the central diagonal is exactly meters and approximately meters.
Answer:

Explanation:
Given the diagonal length, XY, of the end wall of the King’s Chamber of the
Great Pyramid and the length, XZ, of the chamber, find the length, YZ, of the central diagonal is
xy² + xz²  = yz² (using pythagorean theorem)
(7.8)² + (10.5)² = yz²,
yz² = 60.84 +110.25,
yz² = 171.09, square root of 171.09 is 13.08,
yz = 13.08,
So, the length of the cenetral diagonal is exactly 13.08 meters
and approximately 13 meters.

Math in Focus Course 3B Practice 7.3 Answer Key

For this practice, you may use a calculator. Use 3.14 as an approximation for π.
Round your answer to the nearest tenth where necessary.

For each solid, find the value of the variable.

Question 1.

Answer:
The value of variable w is 8 in,

Explanation:
We solid cube have w² + 6 ² = 10² (using pythagorean theorem)
w² = 100 -36,
w² = 64, the value of w is square root of 64 which is 8,
therefore w is 8 in.

Question 2.

Answer:
The value of variable x is 5,

Explanation:
We have x² = 3² + 4² (using pythagorean theorem)
x² = 9 + 16 = 25, the value of x is square root of  25 is 5.

Question 3.

Answer:
The value of variable y is 16 in,

Explanation:
We have 20² = y² + 12² (using pythagorean theorem)
400 = y² + 144,
y² = 400-144 = 256,
so the value of y is sqaure root of 256 is 16,  y = 16 in.

Question 4.

Answer:
The value of variable z is 20 in,

Explanation:
The value of variable z = 2 X 10in = 20 in.

Solve. Show your work. Round your answer to the nearest tenth.

Question 5.
Find the lateral surface area.

Answer:
The lateral surface area is 272 cm²,

Explanation:
Given square pyramid, the lateral surface area(L.A) is 2bh,
where b is base and h is slant height so L.A = 2 × 8cm × 17 cm = 272 cm².

Question 6.
The area of the lateral surface is πrl, where l is the slant height of the cone.
Find the lateral surface area of the cone.

Answer:
The lateral surface of cone is 137.375 cm²,

Explanation:
Given height as 12 cm and slant height l of cone as 12.5 cm so radius of cone is
r ² + 12² = 12.5² (using pythagorean theorem)
r² = 156.25 -144 = 12.25, radius r is square root of 12.25 is 3.5 cm,
now the lateral surface area (L.A) of the cone is  πrl,
L.A = 3.14 × 3.5 × 12.5 = 137.375 cm².

Question 7.
A straw that is 16 centimeters long fits inside the glass shown.
The height of the glass is 14 centimeters. Find the radius of the glass.

Answer:
The radius of the glass is 7.745 approximately 8,

Explanation:
Given a straw that is 16 centimeters long fits inside the glass shown.
The height of the glass is 14 centimeters.
The radius of the glass is 16² = 14² + r² (using pythagorean theorem)
r² = 256- 196 = 60, so r is square root of 60 is 7.745 approximately 8.

Question 8.
A spider sits in a corner of a tank shaped like a rectangular prism. The tank is 13 inches long, 6 inches wide, and 8 inches high. The spider starts to make a web by spinning a length of silk that stretches tightly from one corner along a central diagonal to the opposite corner.

a) Find the length of the diagonal of the rectangular floor of the tank.
Answer:
14.318 is the length of the diagonal of the rectangular floor of the tank,

Explanation:
Given a spider sits in a corner of a tank shaped like a rectangular prism.
The tank is 13 inches long, 6 inches wide, and 8 inches high.
The length of the diagonal of the rectangular floor of the tank is
d² = 13² + 6² = 169 + 36 = 205 so d is square root of 205 is 14.318 in.

b) Find the length of the silk the spider spins from one corner of the tank to the other corner.
Answer:
16.401 in is the length of the silk the spider spins from one corner of the tank to the other corner,

Explanation:
Now the length l of the silk the spider spins from one corner of the tank to the other corner is
l² = 14.318² + 8 ² =  205 + 64 = 269,
so l is square root of 269 is 16.401 in.

Question 9.
A conical party hat ¡s made from a piece of paper as shown. Given that its radius is 3 inches, find the height of the party hat.

Answer:
The height of the party hat is 4 in,

Explanation:
Given a conical party hat ¡s made from a piece of paper is shown with hypotenues 5 in and  its radius is 3 inches,  let the height of the party hat be h so
5² = 3² + h² (using pythagorean theorem)
h² = 25 – 9 = 16, so h is square root of 16 is 4 in.

Question 10.
Mindy wants to make a metal paperweight in the shape of a square pyramid. The paperweight will have the dimensions shown.
a) Find the length of a diagonal of the square base.
Answer:

The length of a diagonal of the square base is 4.24 in,

Explanation:
The length of a diagonal of the square base is
l² = 3² + 3² = 9 + 9 = 18,
so the length is square root of 18 is 4.24 in.

b) Find the height of the paperweight.
Answer:
4 in is the height of the paperweight,

Explanation:
Let h be the height of the paperweight so applying
h² + 3² = 5² (using pythagorean theorem),
h² = 25 – 9 = 16, so h is square root of 16 is 4 in.

Question 11.
Math Journal A box shaped like a rectangular prism is 14.5 centimeters long, 4 centimeters wide, and 3.5 centimeters high. You have a ruler that is 15 centimeters long and 3 centimeters wide. Can it fit inside this box? Explain.
Answer:
No, the ruler cannot fit inside the box,

Explanation:
Given a box shaped like a rectangular prism is 14.5 centimeters long,
4 centimeters wide, and 3.5 centimeters high and I have a ruler that is
15 centimeters long and 3 centimeters wide so by seeing the
length of the ruler is 15 cm which is 15-14.5 = 0.5 more than the rectangular prism,
therefore the ruler cannot fit inside the box.

Practice the problems of Math in Focus Grade 8 Workbook Answer Key Chapter 7 Lesson 7.2 Understanding the Distance Formula to score better marks in the exam.

Math in Focus Grade 7 Course 3 B Chapter 7 Lesson 7.2 Answer Key Understanding the Distance Formula

Math in Focus Grade 8 Chapter 7 Lesson 7.2 Guided Practice Answer Key

Solve.

Question 1.
Points P (-5, 3) and Q (3, -6) are plotted on a coordinate plane. Find the distance between points P and Q. Find both the exact value and an approximate value. Round your answer to the nearest tenth.


So, the distance between points P and Q is exactly units and approximately units.
Answer:
The task is to determine the distance between the points P and Q. compute for the exact vaLue, and then calculate the approximate value.

Note that P is on point (-5, 3) and Q is on point (3, -6), to determine the third vertex R, make a right triangle where Line P and line Q will meet Notice that the vertex wilL be either on (-5, -6) or (3, 3). Say, the third vertex is (-5, -6), then to find the lengths of side \(\overline{P R}\) and \(\overline{Q R}\), determine the absolute value of the y-coordinates of point P and point R, and point Q and point R.
PR = |3 – (-6)|
= |9|
= 9 units
\(\overline{Q R}\) = |3 – (-5)|
= |8|
= 8 units
Apply the Pythagorean Theorem which states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. Note that \(\overline{P R}\) and \(\overline{Q R}\) are the legs of the triangle ∆PQR and Let \(\overline{P Q}\) be the hypotenuse.
PR² + QR² = \(\overline{P Q}^{2}\)   Apply Pythagorean Theorem.
9² + 8^{2 =}  \(\overline{P Q}^{2}\)    Substitution.
81 + 64 = \(\overline{P Q}^{2}\)  Evaluate the Squares.
145 = \(\overline{P Q}^{2}\)   Add.
\(\sqrt{145}=\sqrt{\overline{P Q}^{2}}\) Use Square Root Property.
\(\overline{P Q}\) ≈ 12 Simplify.
Therefore, point P is exactly \(\sqrt{145}\) units away from point Q or around 12 units.

Solve.

Question 2.
Mrs. Smith gives the class the coordinates D (-1, -2), E (2, 4), and F (5, -1) and asks them to join the three points to form triangle DEF.

a) Find the length of \(\overline{\mathrm{DE}}\), \(\overline{\mathrm{EF}}\), and \(\overline{\mathrm{DF}}\).

First find the length of \(\overline{\mathrm{DE}}\).
Let D (-1, -2) be (x₁, y₁) and E(2, 4) be (x₂, y₂).

Answer:
The goal of this task is to analyze the graph of ∆DEF where the coordinates of points D, E, and F are (-1, -2), (2, 4), and (5, -1) respectively then
a. Determine the Length of Line \(\overline{D E}\), \(\overline{E F}\) and \(\overline{D F}\), and
b. Verify if ∆DEF is an isosceles triangle

a) The length of the sides of the triangle can be obtained by applying the distance formula which is the formula used in determining the distance between two points. For the measure of \(\overline{D E}\), let D(-1, -2) be (x₁, y₁) and E(2, 4) be (x₂, y₂).
\(\overline{D E}\) = \(\sqrt{\left(x_{2}-x_{1}\right)+\left(y_{2}-y_{1}\right)}\)   Formula.
= \(\sqrt{(2-(-1))^{2}+(4-(-2))^{2}}\)    Substitution.
= \(\sqrt{3^{2}+6^{2}}\)     Subtract.
= \(\sqrt{9+36}\)     Multiply.
= \(\sqrt{45}\)    Add.
≈ 6.7 Simplify
Therefore,  measures exactly \(\sqrt{45}\) units or approximately 6.7 units.

Next find the length of \(\overline{\mathrm{EF}}\)
Let E (2, 4) be (x₁, y₁) and F(5, -1) be (x₂, y₂).

Answer:
Next, the length of \(\overline{E F}\) can be obtained by applying the distance formula which is the formula used in determining the distance between two points Forthe measure of \(\overline{E F}\), let E (2, 4) be (x₁, y₁) and F(5, -1) be (x₂, y₂)
\(\overline{E F}\) = \(\sqrt{\left(x_{2}-x_{1}\right)+\left(y_{2}-y_{1}\right)}\)   Formula.
= \(\sqrt{(5-2)^{2}+(5-4)^{2}}\)   Substitution.
= \(\sqrt{3^{2}+1^{2}}\)    Subtract.
= \(\sqrt{9+1}\)   Multiply.
= \(\sqrt{10}\)    Add.
≈ 3.2 Simplify
Therefore, \(\overline{E F}\) measures exactly \(\sqrt{10}\) units or approximately 3.2 units.

If two of the three sides of a triangle are of the same length, the triangle is an isosceles triangle.

Then find the length of \(\overline{\mathrm{DF}}\).
Let D (-1, -2) be (x₁, y₁) and F(5, -1) be (x₂, y₂).

Answer:
Lastly, the length of \(\overline{D F}\) can be obtained by applying the distance formula which is the formula used in determining the distance between two points For the measure of \(\overline{D E}\), let D(-1, -2) be (x₁, y₁) and F(5, -1) be (x₂, y₂).
\(\overline{D F}\) = \(\sqrt{\left(x_{2}-x_{1}\right)+\left(y_{2}-y_{1}\right)}\) Formula.
= \(\sqrt{(5-(-1))^{2}+((-1)-(-2))^{2}}\)     Substitution.
= \(\sqrt{6^{2}+1^{2}}\)     Subtract.
= \(\sqrt{36+1}\)     Multiply.
= \(\sqrt{37}\)     Add.
≈ 6.1 Simplify
Therefore, \(\overline{D F}\) measures exactly \(\sqrt{37}\)  units or approximateLy 6.1 units.

b) Is triangle DEF an isosceles triangle? Explain.
Because , triangle DEF is .
Answer:
Note that an isosceles triangle is a triangle whose 2 sides are equal. Since no two sides have the same length, the triangle is not an isosceles triangle.

Solve

Question 3.
Mr. Jacobs gives his students the coordinates R (-3, 3), S(-1, -3), and T(-5, -4) and asks them to join the three points to form triangle RST.

a) Find the lengths of \(\overline{R T}\), \(\overline{S T}\) and \(\overline{R S}\).
First find the length of \(\overline{R T}\).
Let R (-3, 3) be (x₁, y₁) and T(-5, -4) be (x₂, y₂).


The length of \(\overline{R S}\) is units.
Answer:
The task is to analyze the graph of ∆RST where the coordinates of points R, S, and T are (-3, 3), (-1, -2), and (-5, -4) respectively then complete the solution to a. Determine the Length of Line \(\overline{R T}\), \(\overline{S T}\) and \(\overline{R S}\), and b. Verify if ∆DEF is an isosceles triangle.

a) The length of the sides of the triangle can be obtained by applying the distance formula which is the formula used in determining the distance between two points. For the measure of \(\overline{R T}\) substitute R(-3, 3) to (x₁, y₁) while T(-5, -4) to (x₂, y₂). Then,

Since there is no negative magnitude(distance), take only the positive vaLue in ±7.3. Therefore, \(\overline{R T}\) is around 7.3 units.

Next, to determine the Length of \(\overline{S T}\), substitute S(-1, -2) to (x₁, y₁) whiLe T(-5, -4) to (x₂, y₂). Then, applying the distance formula,

Since there is no negative magnitude(distance), take only the positive value in ±4.1 Therefore, \(\overline{S T}\) is approximately 4.1 units.

Lastly, to determine the length of \(\overline{R S}\), substitute R(-3, 3) to (x₁, y₁) while T(-1, -2) to (x₂, y₂). Then, applying the distance formula,

Since there is no negative magnitude(distance), take only the positive value in ±6.3. Therefore, \(\overline{R S}\) is approximately 6.3 units.

b) Is triangle RST an isosceles triangle? Explain.
Because , triangle RST is .
Answer:
Note that an isosceles triangle is a triangle whose 2 sides are equal. Since no two sides have the same length, the triangle is not an isosceles triangle.

Math in Focus Course 3B Practice 7.2 Answer Key

For this practice, you may use a calculator. Round your answers to the nearest tenth if necessary.

Question 1.
Points M (-3, -2) and N (4, 5) are plotted on a coordinate plane. Find the exact distance between points M and N.

Answer:
The goal of this task is to give the exact distance of point M to point N if M has a coordinates (-3, -2) and N is on point (4, 5).

First identify the third vertex forming a right triangle where \(\overline{M N}\) is the hypotenuse. Say Point P is the Last vertex and it is on the point (-3, 5) then find the Lengths of \(\overline{P M}\)and \(\overline{P N}\)
\(\overline{P M}\) = |5 – (-2)|
= 7 units
\(\overline{P N}\) = |-3 – 4|
= 7 units
Apply the Pythagorean Theorem which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two sides then
\(\overline{P M}^{2}+\overline{P N}^{2}=\overline{M N}^{2}\)    Use the Pythagorean Theorem.
7² + 7² = \(\overline{M N}^{2}\)   Substitute values for \(\overline{P M}\) and \(\overline{P N}\)
49 + 49 = \(\overline{M N}^{2}\)    Multiply
98 = \(\overline{M N}^{2}\)   Add.
\(\overline{M N}\) = \(\sqrt{98}\)    Find the positive square root.
\(\overline{M N}\) ≈ 9.9 units   Round to the nearest tenth.
Therefore, the exact distance of point M to point N is \(\sqrt{98}\).

Question 2.
Find the distance between each pair of points. Which pair of points are the greatest distance apart?
a) A(1, 4), B(6, 6)
Answer:
To find the distance of two points, use the distance formula which is the formula used in determining the distance between two points. Then, let A(1, 4) be (x₁, y₁) while B(6, 6) be (x₂, y₁) and g be the distance of A to B.
g = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{(6-1)^{2}+(6-4)^{2}}\)       Substitution.
= \(\sqrt{5^{2}+2^{2}}\)       Subtract.
= \(\sqrt{25+4}\)      Evaluate the squares.
= \(\sqrt{29}\)     Add.
≈ ±5.4    Evaluate.
Note that there is no negative magnitude (distance), thus take only the positive value of g. Therefore, A is 5.4 units away from B.

b) C(-4, 7), D(3, 2)
Answer:
Apply the distance formula then let C(-4, 7) be (x₁, y₁) while D(3, 2) be (x₂, y₂) and h be the distance of C to D.
h = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)   Distance Formula.
= \(\sqrt{(3-(-4))^{2}+(2-7)^{2}}\)    Substitution.
= \(\sqrt{7^{2}+(-5)^{2}}\)   Subtract.
= \(\sqrt{49+25}\)   Evaluate the squares.
= \(\sqrt{74}\)   Add.
≈ ±8.6     Evaluate.
Since there is no negative magnitude(distance), take only the positive value of h. Therefore, the distance of C to D is 8.6 units.

c) L(2, -5), N(-3, -1)
Answer:
Use the distance formuLa then let L(2, -5) be (x₁, y₁) while N(-3, -1) be (x₂, y₂) and o be the distance of L to N.
o = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)   Distance Formula.
= \(\sqrt{((-3)-2)^{2}+((-1)-(-5))^{2}}\)    Substitution.
= \(\sqrt{(-5)^{2}+4^{2}}\)   Subtract.
= \(\sqrt{25+16}\)   Evaluate the squares.
= \(\sqrt{41}\)   Add.
≈ ±6.4      Evaluate.
No magnitude(distance) is negative so take only the positive value of o. Therefore, L is 6.4 units away from N.

d) Y(-6, -4), Z(0, 5)
Answer:
Let Y(-6, -4) be (x₁, y₁) white Z(0. 5) be (x₂, y₂) and p be the distance of Y to Z then apply the distance formula.
p = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)   Distance Formula.
= \(\sqrt{(0-(-6))^{2}+(5-(-4))^{2}}\)    Substitution.
= \(\sqrt{6^{2}+9^{2}}\)   Subtract.
= \(\sqrt{36+81}\)   Evaluate the squares.
= \(\sqrt{117}\)   Add.
≈ ±10.8     Evaluate.
Take only the positive value of p since there is no negative magnitude(distance). Therefore, Y is 10.8 units away from Z.

Question 3.
Zack plots the points R(1, -4), S(-5, 0), and T(5, 2) on a coordinate plane. He joins the three points to form triangle RST. Is the triangle an isosceles triangle? Explain.

Answer:
The target of this task is to verify if ∆RST is an isosceles triangle when points R, S, and T have coordinates (1, -4), (-5, 0) and (5, 2) respectively.

An isosce[es triangLe is a triangle where two of its sides are equal. Calculate the length of \(\overline{R S}\), \(\overline{S T}\), and \(\overline{R T}\) to determine if there exist two sides that are equal by applying the distance formula which is the formula used in determining the distance between two points. For \(\overline{R S}\) let R(1, -4) be (x₁, y₁) while S(-5, 0) be (x₂, y₂) and n be the distance of R to S.
n = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)   Distance Formula.
= \(\sqrt{((-5)-1)^{2}+(0-(-4))^{2}}\)    Substitution.
= \(\sqrt{(-6)^{2}+(-4)^{2}}\)   Subtract.
= \(\sqrt{36+16}\)   Evaluate the squares.
= \(\sqrt{52}\)   Add.
≈ ±7.2     Evaluate.
Note that there is no negative magnitude(distance), thus take only the positive value of n. Therefore, R is 7.2 units away from S.

For \(\overline{S T}\) let S(-5, 0) be (x₁, y₁) while T(5, 2) be (x₂, y₂) and a be the distance of S to T.
a = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)   Distance Formula.
= \(\sqrt{(5-(-5))^{2}+(2-0)^{2}}\)    Substitution.
= \(\sqrt{10^{2}+2^{2}}\)   Subtract.
= \(\sqrt{100+4}\)   Evaluate the squares.
= \(\sqrt{104}\)   Add.
≈ ±10.2     Evaluate.
Since there is no negative magnitude(distance), take only the positive value of a. Therefore, the distance of S to T is 10.2 units.

Then for \(\overline{R T}\) let R(1, -4) be (x₁, y₁) whiLe T(5, 2) be (x₂, y₂) and k be the distance of Sto T.
k = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)   Distance Formula.
= \(\sqrt{(5-1)^{2}+(2-(-4))^{2}}\)    Substitution.
= \(\sqrt{4^{2}+6^{2}}\)   Subtract.
= \(\sqrt{16+36}\)   Evaluate the squares.
= \(\sqrt{52}\)   Add.
≈ ±7.2     Evaluate.
No magnitude(distance) is negative so take only the positive value of k. Therefore, R is 7.2 units away from T.

Notice that \(\overline{R S}\) is 7.2 units, \(\overline{S T}\) is 10.2 units and \(\overline{R T}\) is also 7.2 units. Two sides of ∆RST is equal which is a property of an isosceles triangle. Therefore the triangle is an isosceles triangle.

Use the data in the diagram for questions 4 to 6. Each unit on the grid equals 1 kilometer.

Question 4.
Find the approximate distance from the school to each of the following locations.
a) Zoo
b) Park
c) Swimming pool
d) Library
Answer:
The goal is to determine the estimated distance of school at point A(-2, 1) to the a. Zoo at B(-4, 4), b. Park at C(2, 3), c. Swimming Pool at D(0, -2), and d. Library at E(-5, -3). Note that every unit is equal to 1 kilometer.

a) Calculate for the distance two points A and B to determine the distance of the school and the zoo. Apply the distance formula which is the formula used in determining the distance between two points. Let e be the distance of A to B, and A(-2, 1) be (x₁, y₁) while B(-4, 4) be (x₂, y₂).
e = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{((-4)-(-2))^{2}+(4-1)^{2}}\)      Substitution.
= \(\sqrt{(-2)^{2}+(3)^{2}}\)     Subtract.
= \(\sqrt{4+9}\)    Evaluate the squares.
= \(\sqrt{13}\)    Add.
≈ ±3.6    Evaluate.
Because that there is no negative magnitude(distance), take only the positive value of e. Therefore, the zoo is around 3.6 kilometers away from the school.

b) Compute for the distance of points A and C to know the distance of school to park by applying the distance formula describe in (a). Then Let f be the distance of A to C, and A(-2, 1)) be (x₁, y₁) whiLe C(2, 3) be (x₂, y₂).
f = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{(2-(-2))^{2}+(3-1)^{2}}\)      Substitution.
= \(\sqrt{4^{2}+2^{2}}\)     Subtract.
= \(\sqrt{16+4}\)    Evaluate the squares.
= \(\sqrt{20}\)    Add.
≈ ±4.5    Evaluate.
Since there is no negative magnitude(distance), take only the positive value of f. Therefore, the distance of school to the park is approximately 4.5 kiLometers.

c) Solve for the distance of two points A and D to identify the distance of the school and the swimming pool using the distance formula as mentioned above. Next, Let g be the distance of A to D, and A(-2, 1)) be (x₁, y₁) while D(0, -2) be (x₂, y₂).
g = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{(0-(-2))^{2}+((-2)-1)^{2}}\)      Substitution.
= \(\sqrt{2^{2}+(-3)^{2}}\)     Subtract.
= \(\sqrt{4+9}\)    Evaluate the squares.
= \(\sqrt{13}\)    Add.
≈ ±3.6      Evaluate.
No magnitude(distance) is negative thus take only the positive value of g. Therefore, the swimming pool is around 3.6 kilometers away from the school.

d) Calculate for the distance of point A and point E to determine the distance ci me school and the library by applying the distance formula Let h be the distance of A to E, and A(-2, 1)) be (x₁, y₁) while E(-5, -3) be (x₂, y₂).
h = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{((-5)-(-2))^{2}+((-3)-1)^{2}}\)      Substitution.
= \(\sqrt{(-3)^{2}+(-4)^{2}}\)     Subtract.
= \(\sqrt{9+16}\)    Evaluate the squares.
= \(\sqrt{25}\)    Add.
≈ ±5      Evaluate.
There is no negative magnitude(distance) so take only the positive value of h. Therefore, the library is 5 kilometers away from the school.

Question 5.
Which two locations are the same distance from the school?
Answer:
The objective of this task is to determine the locations that are equal in distance from the school.

Note that by applying the distance formula which is the formula used in determining the distance between two points. The distance of A(school) to B(park) is computed by substituting A(-2, 1) to (x₁, y₁) while B(-4, 4) to (x₂, y₂) and letting e be the distance.
e = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{((-4)-(-2))^{2}+(4-1)^{2}}\)      Substitution.
= \(\sqrt{(-2)^{2}+(3)^{2}}\)     Subtract.
= \(\sqrt{4+9}\)    Evaluate the squares.
= \(\sqrt{13}\)    Add.
≈ ±3.6     Evaluate.
Thus, the park is 3.6 kilometers from the school.

Also, A(-2, 1) is substituted to (x₁, y₁) whiLe D(0, -2) to (x₂, y₂) to calculate for the distance of the school to the swimming pool.
g = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{(0-(-2))^{2}+((-2)-1)^{2}}\)      Substitution.
= \(\sqrt{2^{2}+(-3)^{2}}\)     Subtract.
= \(\sqrt{4+9}\)    Evaluate the squares.
= \(\sqrt{13}\)    Add.
≈ ±3.6       Evaluate.
Thus, the swimming pool is 3.6 kilometers from the school.
Therefore, the swimming pool and the park is have the same distance from the schooL

Question 6.
Which location is farthest from the school?
Answer:
The task is to identify the location with the largest distance from the school at point A(-2, 1) if the zoo is at B( -4, 4), park at C(2, 3), swimming pool at D(0, -2), and Library at E(-5, -3).

First calculate for the distance of points A and B to determine the distance of the school and the zoo. Apply the distance formula which is the formula used in determining the distance between two points. Let h be the distance of A to B, and A(-2, 1) be (x₁, y₁) while B(-4, 4) be (x₂, y₂).
h = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{((-4)-(-2))^{2}+(4-1)^{2}}\)      Substitution.
= \(\sqrt{(-2)^{2}+(3)^{2}}\)     Subtract.
= \(\sqrt{4+9}\)    Evaluate the squares.
= \(\sqrt{13}\)    Add.
≈ ±3.6       Evaluate.
Because that there is no negative magnitude(distance), take only the positive vaLue of h. Therefore, the zoo is around 3.6 kilometers away from the school.

Next, compute for the distance of points A and C to know the distance of school to park by app[ying the distance formula describe in (a). Then let j be the distance of A to C, and A(-2, 1) be (x₁, y₁) while C(2, 3) be (x₂, y(x₁, y₁)).
j = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{(2-(-2))^{2}+(3-1)^{2}}\)      Substitution.
= \(\sqrt{4^{2}+2^{2}}\)     Subtract.
= \(\sqrt{16+4}\)    Evaluate the squares.
= \(\sqrt{20}\)    Add.
≈ ±4.5       Evaluate.
Since there is no negative magnitude(distance), take only the positive vaLue of j Therefore, the distance of the school to the park is approximately 4.5 kiLometers.

Now, solve for the distance of two points A and D to identify the distance of schooL and the swimming pool using the distance formula as mentioned above. Let k be the distance of A to D, and A(-2, 1) be (x₁, y₁) while D(0, -2) be (x₂, y₂).
k = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{(0-(-2))^{2}+((-2)-1)^{2}}\)      Substitution.
= \(\sqrt{2^{2}+(-3)^{2}}\)     Subtract.
= \(\sqrt{4+9}\)    Evaluate the squares.
= \(\sqrt{13}\)    Add.
≈ ±3.6      Evaluate.
No magnitude(distance is negative thus take only the positive value of k. Therefore, the swimming pool is around 3.6 kilometers away from the school.

Lastly, calculate for the distance of point A and point E to determine the distance of the schooL and the library by applying the distance formula Let l be the distance of A to E, and A(-2, 1) be (x₁, y₁) whiLe E(-5, -3) be (x₂, y₂).
l = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{((-5)-(-2))^{2}+((-3)-1)^{2}}\)      Substitution.
= \(\sqrt{(-3)^{2}+(-4)^{2}}\)     Subtract.
= \(\sqrt{9+16}\)    Evaluate the squares.
= \(\sqrt{25}\)    Add.
= ±5    Evaluate.
There is no negative magnitude(distance) so take only the positive value of l. Therefore, the library is 5 kilometers away from the school.
Since the school has a distance of 3.6 kilometers for both the zoo and swimming pool, has a distance of 4.5 kilometers to the park and 5 kilometers to the Library. Therefore, library has the farthest distance from the school.

Solve. Show your work.

Question 7.
A ship is located at the point shown in the diagram. The ship needs to stop at a port for refueling on its way to the lighthouse. It can either stop at Port A or Port B. If the captain wants the journey to be as short as possible, which port should the ship stop at?

Answer:
The goal is to find which port the captain should pass for a shorter journey to the Lighthouse at (4, 4) if the ship is on point (-4, -2) and the two possible port to stop are Port A(-3, 3) and Port B(3, 1)

First, calculate the distance or me ship to Port A to the from Port A to the Lighthouse. Afterward, add the computed distances to determine the total distance the ship will travel if it will pass through Port A. Applying the distance formula which is the formula used in determining the distance between two points. Let the position of the ship (-4, -2) be (x₁, y₁) whiLe Port A(-3, 3) be (x₂, y₂) and u denoted their distance.
u = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{((-3)-(-4))^{2}+(3-(-2))^{2}}\)      Substitution.
= \(\sqrt{1^{2}+5^{2}}\)     Subtract.
= \(\sqrt{1+25}\)    Evaluate the squares.
= \(\sqrt{26}\)    Add.
≈ ±5.1    Evaluate.
Note that there is no negative magnitude(distance) so only take u equals to 5.1. Thus, the ship is 5.1 units away from Port A.
For the distance of Port A to the [ighthouse, let Port A(-3. 3) be substituted to (x₁, y₁) while the position (4, 4) of the Lighthouse to (x₂, y₂) and let w be their distance then
w = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{(4-(-3))^{2}+(4-3)^{2}}\)      Substitution.
= \(\sqrt{7^{2}+1^{2}}\)     Subtract.
= \(\sqrt{49+1}\)    Evaluate the squares.
= \(\sqrt{50}\)    Add.
≈ ±7.1    Evaluate.
Hence, Port A and the Lighthouse has a distance of 7.1 units. Now add the two distances u and w,
u + w = 5.1 – 7.1
= 12.2 units
Thus, the total distance that the ship will travel if it stops to Port A then proceed to the Lighthouse is 12.2 units.

Next, calculate the distance of the ship to Port B then from Port B to the Lighthouse. Then, add the computed distances to determine the total distance the ship will travel if it will pass through Port B. Apply the distance formula to compute for the distance of the ship to the Port B. Let the position of the ship (-4, -2) be (x₁, y₁) while Port B(3, 1) be (x₂, y₂) and y denoted their distance.
y = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{(3-(-4))^{2}+(1-(-2))^{2}}\)      Substitution.
= \(\sqrt{7^{2}+3^{2}}\)     Subtract.
= \(\sqrt{49+9}\)    Evaluate the squares.
= \(\sqrt{58}\)    Add.
≈ ±7.6      Evaluate.
Thus, the ship is 7.6 units away from Port B.

For the distance of Port B to the [ighthouse, [et Port B(3, 1) be substituted to (x₁, y₁) white the position (4, 4) of the lighthouse to (x₂, y₂) and let z be their distance then
z = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)    Distance Formula.
= \(\sqrt{(4-3)^{2}+(4-1)^{2}}\)      Substitution.
= \(\sqrt{1^{2}+3^{2}}\)     Subtract.
= \(\sqrt{1+9}\)    Evaluate the squares.
= \(\sqrt{10}\)    Add.
≈ ±3.2       Evaluate.
Hence, Port B and the lighthouse has a distance of 3.2 units. Now add the two distances y and z,
y + z = 7.6 + 3.2
= 10.8 units
Thus, the total distance that the ship will travel if it stop to Port B then proceed to the lighthouse is 10.8 units.
Therefore, the ship should take the route passing to Port B to the lighthouse for a shorter journey.

Question 8.
Blair drew two triangles on a coordinate plane.
a) Find the lengths of the sides of triangle XYZ Classify the triangle by its side lengths.
Answer:

b) Blair thinks that triangle RST is a right triangle. Do you agree? Why or why not?
Answer:
Determine the sides of the triangle by applying the distance formula. Afterwards, use the Converse of Pythagorean Theorem which states that if the square of the length of the hypotenuse of a triangle is equal to the sum of the squares of the lengths of the two legs, then the triangle is a right triangle. If the hypotenuse of ∆RST is equal to the sum of the square of its legs then ∆RST is indeed a right triangle. To calculate side \(\overline{S R}\), substitute S(1, 4) to (x₁, y₁) while R(4, 5) to (x₂, y₂)


Thus, \(\overline{R T}\) is 5.1 units Long. Since \(\overline{R T}\) has the largest length use it as the hypotenuse then
(5.1)² = (45)² + (3.2)²  Use the Converse Pythagorean Theorem.
26.01 = 20.25 + 10.24    Evaluate the squares.
26.01 ≠ 30.49   Add.
Since, the square of \(\overline{R T}\) is not equal to the sum of the squares of ST and SR, then ∆RST is not a right triangle.

Question 9.
The engine on a cruise ship has broken down and the passengers are waiting for a patrol ship to rescue them. The positions of the two ships are shown on the right. Each unit on the grid equals 6 miles.

a) How far is the patrol ship from the cruise ship?
Answer:

The objective of this task is to analyze the positions of the cruise ship and the patrol ship then a. Find the distance of the two ships to each other, and b. Determine how long will it takes the patrol ship to reach the boat if its traveling at the speed of 50 miles per hour.

Observe that position of the patrol ship denoted by point S is on (-4, 5) while the coordinates of the cruise ship represented by point H is (2, -3) Say vertex P is on (2, 5) and forms a right triangle with the other two points then,
\(\overline{S P}\) = |-4 – 2|
= |-6|
= 6 units
\(\overline{H P}\) = |-3 – 5|
= |-8|
= 8 units
Apply the Pythagorean Theorem which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two other sides then
\(\overline{S P}^{2}+\overline{H P}^{2}=\overline{S H}^{2}\)    Use the Pythagorean Theorem.
6² + 8² = \(\overline{S H}^{2}\)    Substitute values for \(\overline{S P}\) and \(\overline{H P}\)
36 + 64 = \(\overline{S H}^{2}\)    Multiply.
100 = \(\overline{S H}^{2}\)      Add.
\(\overline{S H}\) = \(\sqrt{100}\)    Find the positive square root.
\(\overline{S H}\) = 10 units.    Simplify.
Since each unit is equal to 6 miles then multiply 10 units to 6. Therefore, the patrol ship is 60 miles away from the other ship.

b) If the patrol ship travels at a speed of 50 miles per hour, how long will it take to reach the boat?
Answer:
Since the patrol ship is 60 miles away from the cruise ship and it is travelling at the speed of 5Omi[es per hour, then divide this speed to the distance of the ships to each other to know how long will it takes the patrol ship to reach the boat.
= \(\frac{60 \text { miles }}{50 \text { miles } / \text { hour }}\)
1.2 hour
Therefore, the patrol ship will meet the other ship in 1.2 hours or in 1 hour and 12 minutes.

Question 10.
Math Journal Point P (1,2) is the center of a circle. Three of the following points lie on the circle. Tell which point is not on the circle. Explain your reasoning. A (-1, 4), B (4, 4), C (3, 0), and D (3, 4)
Answer:
The goal of this task is to find out which point among D(3, 4), C(3, 0), B(4, 4), and A(-1, 4) does not belong on circle P whose center is at (1, 2).

To determine which points does note belong to the circle, solve for the distance between the center of the circle and each of the points by applying the distance formula. This states that distance is equal to the square root of the sum of the x-coordinate of the second point minus the x-coordinate of the first point and the y-coordinate of the second point minus the y-coordinate of the first point. Let (1, 2) be (x₁, y₁) and (3, 4) be (x₂, y₂) and p be the distance of the center from point D, then
p = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)   Formula.
= \(\sqrt{(3-1)^{2}+(4-2)^{2}}\)       Substitution.
= \(\sqrt{2^{2}+2^{2}}\)       Subtract.
= \(\sqrt{4+4}\)       Evaluate the squares.
= \(\sqrt{8}\)       Add.
≈ ±2.8          Evaluate.
Therefore, the center of the circle is approximateLy 2.8 units from point D.

Applying the distance formula, let (1, 2) be (x₁, y₁) and (3, 0) be (x₂, y₂) and q be the distance of the center from point C, then
q = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)   Formula.
= \(\sqrt{(3-1)^{2}+(0-2)^{2}}\)       Substitution.
= \(\sqrt{2^{2}+(-2)^{2}}\)       Subtract.
= \(\sqrt{4+4}\)       Evaluate the squares.
= \(\sqrt{8}\)       Add.
≈ ±2.8 Evaluate.
Therefore, the center of the circle is around 2.8 units from point C.

Next, using the distance formula, let (1, 2) be (x₁, y₁) and (4, 4) be (x₂, y₂) and r be the distance of the center from point B, then
r = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)   Formula.
= \(\sqrt{(4-1)^{2}+(4-2)^{2}}\)       Substitution.
= \(\sqrt{3^{2}+2^{2}}\)       Subtract.
= \(\sqrt{9+4}\)       Evaluate the squares.
= \(\sqrt{13}\)       Add.
≈ ±3.6         Evaluate.
Therefore, the center of the circle is about 3.6 units from point B.

Applying the distance formula for the last time, let (1, 2) be (x₁, y₁) and (-1, 4) be (x₂, y₂) and s be the distance of the center from point A, then
s = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)   Formula.
= \(\sqrt{(-1)-1)^{2}+(4-2)^{2}}\)       Substitution.
= \(\sqrt{(-2)^{2}+2^{2}}\)       Subtract.
= \(\sqrt{4+4}\)       Evaluate the squares.
= \(\sqrt{8}\)       Add.
≈ ±2.8     Evaluate.
Therefore, the center of the circle is approximateLy 2.8 units from point A. Notice that points A, C and D has the same distance from the center of the circle. Thus point B does not belong on circle P.

Question 11.
Math Journal Jose found the distance between points X and Y by first counting grids to find the horizontal and vertical distances. Then he used the Pythagorean Theorem. How is his method different from using the distance formula? How is it the same?
Answer:
The task is to state the similarity and difference of the Pythagorean Theorem and the distance formula.

Both the Pythagorean Theorem and the Distance Formula is utilized to determine the measure of a Line segment The Pythagorean Theorem is used most of the time when the Length of a diagonal Line is unknown The method includes generating Line segments that will serve as the legs of the hypotenuse. This legs must have known measurements so the method will work. The distance formula on the other hand only needs the coordinates of the endpoints of the line so the measure of the length of the segment can be computed.

Practice the problems of Math in Focus Grade 8 Workbook Answer Key Chapter 7 Lesson 7.1 Understanding the Pythagorean Theorem and Plane Figures to score better marks in the exam.

Math in Focus Grade 7 Course 3 B Chapter 7 Lesson 7.1 Answer Key Understanding the Pythagorean Theorem and Plane Figures

Math in Focus Grade 8 Chapter 7 Lesson 7.1 Guided Practice Answer Key

Hands-On Activity

Materials:

• scissors

Use Rearrangementto Prove The Pythagorean Theorem

In a right triangle, the longest side is known as the hypotenuse. The two shorter sides are called legs.

Step 1.
Draw four identical right triangles, with side lengths a, b, and c, on a a piece of paper and cut them out. Make one leg shorter than the other.

Step 2.
Arrange the triangles to form a square with side length c.

Step 3.
What is the area of the square whose sides are c units long? Label the large square with its area.

Step 4.
Draw two squares on a different piece of paper. One square should have side lengths of b units, and the other should have side lengths of a units. You can use one of the triangles as a guide to mark off a units and b units. Cut out these square and arrange the four triangles and the two new squares to form a large square like the one shown.

Step 5.
Write an algebraic expression in terms of a and b for the combined area of the two small squares.

Math Journal Compare the areas of the squares found in Step 3 and Step 5. Write an equation that relates the area of one square to the sum of the areas of the other two squares.

In the activity, you explored the relationship among the sides of a right triangle. This relationship is true for all right triangles and is described by the Pythagorean Theorem.

In the activity, you observed the following:

Pythagorean Theorem
The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs. For the triangle shown, you can write this equation: a² + b² = c².

Complete.

Question 1.
Merlin wants to put a fence around a right triangular garden.
He measures two sides. Find the length of the unknown side.

Answer:
The length of the unknown side is 15 ft,

Explanation:

Complete.

Question 2.
Find the values of x and y. Round your answer to the nearest tenth.

Notice that there are two right triangles.
One has a hypotenuse that is 2.5 inches long, and the other has a hypotenuse that is y inches long.
First find the value of x.

Answer:
The value of x is approximately 2 and
the value of y is approximately 4

Hands-On Activity

Materials

• graph paper
• ruler

Explore Pythagorean Triples
A Pythagorean triple is any set of three whole numbers that satisfy the
Pythagorean Theorem equation c² = a² + b². The table shows several sets of Pythagorean triples.

Step 1.
Select one Pythagorean triple from the table. Draw a triangle with legs a units and b units long on a piece of graph paper.

Step 2.
Using a strip of graph paper as a measuring tool, measure the hypotenuse of the triangle. Is it equal to the greatest value in the Pythagorean triple?

Step 3.
Repeat Step 1 and Step 2 using the other Pythagorean triples in the table.

Step 4.
The converse of the Pythagorean Theorem says that if the sum of the squares of the lengths of the two legs of a triangle equals the square of the length of the hypotenuse, then the triangle is a right triangle. Do your triangle measurements support this statement? Explain.

Copy and complete each with a value and each with = or ≠.

Question 3.
A triangle has side lengths 13 centimeters, 15 centimeters, and 18 centimeter Is it a right triangle?

Answer:
The triangle is a not a right triangle,

Explanation:

Complete.

Question 4.
A ladder 18 feet long is leaning against a wall. The base of the ladder is 9 feet away from the wall. Find the distance from the top of the ladder to the ground. Round your answer to the nearest tenth.
Let the distance from the top of the ladder to the ground be x feet.

The distance from the top of the ladder to the ground is approximately feet.
Answer:
The distance from the top of the ladder to the ground is approximately 16 feet,

Explanation:

Complete.

Question 5.
A tree has a shadow length of approximately 9 feet. The distance from the tip of the tree to the tip of the shadow is about 15 feet. How tall is the tree?

Let the height of the tree be x feet.

Answer:
The height of the tree is 12  feet,

Explanation:

Question 6.
The support pole of the tent shown forms one leg of a right triangle. One side of the tent forms the hypotenuse of the right triangle. Find the length of the base of the tent.

So, the length of the base of the tent is inches.
Answer:
The length of the base of the tent is 48 in,

Explanation:

Math in Focus Course 3B Practice 7.1 Answer Key

For this practice, you may use a calculator.

Copy the road signs shown below, identify a right triangle in each sign,
and label the hypotenuse with an arrow.

Question 1.

Answer:

Explanation:
Identified a right triangle in the sign,
and labeled the hypotenuse with an arrow.

Question 2.

Answer:

Explanation:
Identified a right triangle in the sign,
and labeled the hypotenuse with an arrow.

Find the value of x.

Question 3.

Answer:
x = 15 cm,

Explanation:
Using pythagorean x ² = 9²  + 12² = 81 + 144 =225,
x is equal to square root of 225 which is 15 cm.

Question 4.

Answer:
x = 3.96 approximately 4 cm,

Explanation:
Using pythagorean x ² + 4.5²  = 6² ,
x ² + 20.25 = 36,
x ² = 36 – 20.25 = 15.75,
x is equal to square root of 15.75 which is 3.96 approximately 4 cm.

Question 5.

Answer:
4 cm

Explanation:
Using pythagorean x ² + 10²  = 12.5² ,
x ² + 100 = 156.25,
x ² = 156.25 – 100 = 15.75,
x is equal to square root of 15.75 which is 3.96 approximately 4 cm.

Question 6.

Answer:

Using pythagorean x² + 4.5²  = 6² ,
x ² + 20.25 = 36,
x ² = 36 – 20.25 = 15.75,
x is equal to square root of 15.75 which is 3.96 approximately 4 cm.

Calculate each unknown side length. Round your answer to the nearest tenth.

Question 7.

Answer:
x is 3 inches and y is 12.64 approximately 13,

Explanation:
First we calculate the given right angled triangle using
phythagorean theorem x² + 4² = 5² ,
x² = 25- 16 = 9 squareroot of 9 is 3 therefore
x is 3 inches, Now
3² + y ² = 13² ,
y² = 169- 9 = 160,so y is square root of 160 is 12.64 approximately 13.

Question 8.

Answer:
the unknown side length is 20 in,

Explanation:
We take x as y +z inches,
First right angle triangle as base z and height 16 in and hypotenuse as 20,
so z² + 16² = 20²,
z² = 400 – 256 = 144     z is square root of 144 is 12 inches,
Now anthe triangle base y height 16 in and hypotenuse is 18
so y² + 16² = 18²,
y² = 324- 256 = 68 so y is square root of 68 is 8.24 in,
so x = y + z = 8.24 in + 12 in = 20.24 approximately 20.

Question 9.

Answer:
x is 6 in and y is 12in,

Explanation:
Given smaller right andle triangle and bigger right traingle,
First smaller using pythogorean theorem
x² + 4² = 7² ,
x² = 49-16= 33, x is square root of 33 is 5.744 approximately 6,
Now 10² + 6² = y² ,
y² = 100 + 36 = 136, so y is square root of 136 is 11.66 approximately 12,
So x = 6 in and y = 12 in respectively.

Question 10.

Answer:
x = 4in and y = 9 in,

Explanation:
Given right angled triangle applying pythagorean theorem we have
x² + 8² = 9²,
x² = 81 – 64 = 17, so square root of 17 is 4.123 approximately 4
x = 4 in  and y= 9 in.

Solve. Show your work. Round your answer to the nearest tenth.

Question 11.
Daniel had two pieces of wire. He bent each piece of wire into the shape of a triangle. Determine which triangle is a right triangle.

Answer:
Triangle B is a right triangle,
Explanation:
Given Triangle A  applying pythagorean theorem
26² + 18² = 36² ,
676 + 324 = 1296,
1000 ≠ 1296,
For Triangle B applying pythagorean theorem
20² + 15² = 25²,
400 +  225 = 625,
625 = 625,
The converse of the Pythagorean Theorem says that if the sum of the squares of the lengths of the two legs of a triangle equals the square of the length of the hypotenuse, then the triangle is a right triangle so triangle B is right triangle.

Question 12.
Kendrick wants to build a slide for his son in the backyard. He buys a slide that is 8 feet long. The height of the stairs is 5 feet. Find the distance from the bottom of the stairs to the base of the slide.

Answer:
The distance from the bottom of the stairs to the base of the slide is
approximately 6 ft,

Explanation:
Given
Kendrick wants to build a slide for his son in the backyard.
He buys a slide that is 8 feet long. The height of the stairs is 5 feet.
To find the distance from the bottom of the stairs to the base b of the slide applying pythagorean theorem as 5² + b² = 8²,
b² = 64 – 25 = 39, square root of b is 6.244 approximately 6 ft.

Question 13.
Mrs. Hanson uses a wheelchair. Her husband decides to build a ramp to make it easier for her to enter and leave the house. Find the length of the ramp.

Answer:
The length of the ramp is approximately 71 in,

Explanation:
Given Mrs. Hanson uses a wheelchair. Her husband decides to build a ramp of r inches to make it easier for her to enter and leave the house.
As triangle is right triangle applying pythagorean theorem
r² = 12² + 70²,
r² = 144 + 4900 =5044, r is square root of 5044 is 71.02 in,
So the length of the ramp is approximately 71 in.

Question 14.
A sign is hung outside a shop on the wall of a building. One end of the brace that holds the sign is connected to the wall. The other end of the brace is connected to the wall by a wire. Use the dimensions in the diagram to find the length of the wire.

Answer:
The length of wire is approximately 4 ft,

Explanation:
Given a sign is hung outside a shop on the wall of a building.
One end of the brace that holds the sign is connected to the wall.
The other end of the brace is connected to the wall by a wire.
the length l of the wire is l² = 3² + 3²,
l² = 9 + 9 = 18, l is square root of 18 is 4.24 ,
so the length of wire is approximately 4 ft.

Question 15.
Kelly is flying a kite. When the string, which is 34 feet long, becomes taut, the distance along the ground from the kite to Kelly’s hand is 6.5 feet. Find the vertical height of the kite above her hand.

Answer:
Vertical height of the kite above her hand is 33.37 approximately 34 feet,

Explanation:
Given Kelly is flying a kite. When the string, which is 34 feet long,
becomes taut, the distance along the ground from the kite to Kelly’s hand is 6.5 feet.
Let the vertical height v of the kite above her hand. As it is right triangle applying pythogorean theorem
34² = 6.5² + v²,
v² = 1156 – 42.25 = 1113.75,
so vertical height is square root of 1113.75 is 33.37 approximately 34 feet.

Question 16.
In a movie scene, an actor runs up a ramp that leads from the top of one building to the top of another building. The horizontal distance between the two buildings is 8.6 meters. The height of the shorter building is 18 meters while the height of the taller building is 24 meters. Find the length of the ramp.

Answer:
The length of ramp is is 6.16 approximately 6 m,

Explanation:
Given in a movie scene, an actor runs up a ramp that leads from the top of one building to the top of another building.
The horizontal distance between the two buildings is 8.6 meters.
The height of the shorter building is 18 meters while the height of the taller building is 24 meters. The length of ramp is l and it is right triangle so
l² = 8.6² -(24-18)²,
l² =  73.96 – (6)²,
l² = 73.96 – 36 = 37.96 , so l is square root of 37.96 is 6.16 approximately 6 m.

Question 17.
The size of a rectangular television screen is given as the length of the diagonal of the screen. Find the size of a television screen that has a height of 38.4 inches and width of 28.8 inches.
Answer:
the length of the diagonal of the screen is 48 inches,

Explanation:
Given the size of a rectangular television screen is given as the length l of the diagonal of the screen.
The size of a television screen that has a height of 38.4 inches and width of 28.8 inches is
l² = 38.4² + 28.8²,
l² = 1474.56 + 829.44= 2304, l is the square root of 2304 is 48 inches.

Question 18.
A ship sailed from Port X to Port Y. It traveled 20 kilometers due north and then 25 kilometers due west. Find the shortest distance between the two ports.
Answer:
The shortest distance between the two ports is 32.01 approximately 32 kilometers,

Explanation:
Given a ship sailed from Port X to Port Y. It traveled 20 kilometers due north and then 25 kilometers due west. Let s be the shortest distance between the two ports its right triangle applying pythagorean theorem
s² = 20² + 25²
= 400+ 625
= 1025, s is the square root of 1025 is 32.01
approximately 32 kilometers.

Question 19.
Mike wants to build a sailboat. The scale drawing is shown on the right. Find the values of x and y.

Answer:
The value of x is 7.07 approximately 7 and
y is 14.21 approximately 14,

Explanation:
Given both right traingles 1st triangle applying
pythagorean theorem x² + x² = 10²,
2x² = 100,
x² = 100/2 = 50 so x is square root of 50 is 7.07 approximately 7,
and 2nd triangle applying pythogorean theorem
y² = 11² +( 7+2)²,
y² = 121 + 81 = 202, y is squareroot of 202 is 14.21 approximately 14.

Question 20.
The diagonal length of a square window is 40 centimeters. Find the area of the window.
Answer:
The area of window is 800 cm²,

Explanation:

Question 21.
Triangles ABC and ACD are right triangles. AB is 28 meters long. AC is 21 meters long. Find the lengths of \(\overline{B C}\) and \(\overline{A D}\).

Answer:

Explanation:
Given Triangles ABC and ACD are right triangles. AB is 28 meters long.
AC is 21 meters long.
so BC² = 21² + 28²,
Bc ² = 441 + 784 = 1225, so Bc is square root of 1225 is 35 m.

Question 22.
The infield of a baseball diamond is a square. Barry measures the distance from home plate to second base and finds that it is about 38.7 meters.

a) Find the approximate distance from home plate to first base.
Answer:
The approximate distance from home plate to first base is 27.4 m

Explanation:
The 38.7 meters is the diagonal of a square; the distance from home plate to first base is a side of the square. The length of the side of the square is the length of the diagonal, divided by the square root of 2.
=38.7/ 1.414 =  27.4 meters.

b) Find the area of the infield.
Answer:
The area of the infield is 750.76 square meters,

Explanation:
Given the side of a baseball diamond is a square and side distance is 27.4 m
so the area of the infield is side × side = 27.4 × 27.4 = 750.76 square meters.

Question 23.
The road sign shown is in the shape of an equilateral triangle. The height of the road sign is 15 inches.

a) Find the length of each side.
Answer:
The length of each side is 17 in,

Explanation:
As we know in an equilateral triangle h = (1/2) × √3 × a.
Angles of Equilateral Triangle: A = B = C = 60° ,
Sides of Equilateral Triangle: a = b = c.
So 15 = 1/2 × √3 × a,
30 = 1.732 × a,
a = 30/1.732 = 17.32 approximately 17 in.

b) Find the area of the sign.
Answer:
The area of the sign is 127.5 in²,

Explanation:
As we know area of triangle is 1/2 × base ×  height,
1/2 × 17 in × 15 in = 127.5 in².

Question 24.
Math Journal A salesman selling windows says that a window frame J that is 8 feet long and 6.5 feet wide is rectangular. He also says that it has a diagonal length of 12 feet. Danielle thinks that the salesman is wrong in saying that the window is rectangular. Who is right and why?
Answer:
Yes, Danielle thinks that the salesman is wrong in saying that the window is rectangular,

Explanation:
Given a salesman selling windows says that a window frame J that is 8 feet long and 6.5 feet wide is rectangular.
He also says that it has a diagonal length of 12 feet.
But if we calculate the diagonal length we get

d = 10 not 12 so sales man is wrong.

Question 25.
Math Journal Fiona buys a triangular table. The sides of the table top are 29.4 inches, 39.2 inches, and 49 inches long. She wants to place the table in a corner of a rectangular room. Will the table fit snugly in the corner? Explain.
Answer:
Yes the table fit snugly in the corner,

Explanation:
Given Fiona buys a triangular table.
The sides of the table top are 29.4 inches, 39.2 inches, and 49 inches long.
She wants to place the table in a corner of a rectangular room.
The table fit snugly in the corner.

Practice the problems of Math in Focus Grade 8 Workbook Answer Key Chapter 7 The Pythagorean Theorem to score better marks in the exam.

Math in Focus Grade 8 Course 3 A Chapter 7 Answer Key The Pythagorean Theorem

Math in Focus Grade 8 Chapter 7 Quick Check Answer Key

Find the square of each number.

Question 1.
3
Answer:
9,

Explanation:
Given number is 3 the square is 3² = 3 X 3 = 9.

Question 2.
\(\frac{1}{4}\)
Answer:
\(\frac{1}{16}\),

Explanation:
Given number is \(\frac{1}{4}\) the sqaure is
(\(\frac{1}{4}\))² = \(\frac{1}{4}\) X \(\frac{1}{4}\) =
\(\frac{1 X 1}{4 X 4}\) = \(\frac{1}{16}\).

Question 3.
-7
Answer:
49,

Explanation:
Given number is -7 the square is (-7)² = -7 X -7 = 49.

Find the square roots of each number.

Question 4.
16
Answer:
4,

Explanation:

Question 5.
64
Answer:
8,

Explanation:

Question 6.
400
Answer:
20,

Explanation:

Find the cube of each number.

Question 7.
\(\frac{1}{2}\)
Answer:
\(\frac{1}{8}\),

Explanation:
Given \(\frac{1}{2}\) the cube is (\(\frac{1}{2}\))³ =
\(\frac{1}{2}\) X \(\frac{1}{2}\) X \(\frac{1}{2}\) =
\(\frac{1 X 1 x 1}{2X 2 X 2}\) =\(\frac{1}{8}\).

Question 8.
7
Answer:
343

Explanation:
Given 7 the cube is (7)³ = 7 X 7 X 7 = 343.

Question 9.
11
Answer:
1331,

Explanation:
Given 11 the cube is (11)³ = 11 X 11 X 11 = 1331.

Find the cube root of each number.

Question 10.
8
Answer:
2

Explanation:

Question 11.
27
Answer:
3

Explanation:

Question 12.
125
Answer:
5

Explanation:

Use graph paper. Plot each pair of points on a coordinate plane. Connect the points to form a line segment and find its length.

Question 13.
(3, 5) and (3, 9)
Answer:
The objective is to plot and connect point (-3, 1) and point(-3, -3) on a coordinate plane, then identify the length of the line that the two points create.

Note that to represent point (-3, 1), point out in which part of the graph does -3 and 1 meet. -3 is 3 units to the left of 0 and 1 is 1 unit up. Likewise, for point (-3, -3), point out in which part of the graph does -3 and -3 meet. The first -3 is 3 units to the left of 0 and the other -3 is 3 units down. Then, let O be the point (-3, 1), and P be the point (-3, -3).

To find the length of the line segment \(\overline{O P}\), get the absolute value of the difference of y-coordinate of point O  to the y-coordinate of P since points O and P have the same x-coordinate which is -3. Then
length of \(\overline{O P}\) = |1 – (-3)|
= |1 + 3|
= 4
Therefore, the length of \(\overline{O P}\) is 4 units.

Question 14.
(3, -6) and (3, -9)
Answer:
The goal of this task is to plot and connect point (1, 2) and point (1, -1) on a coordinate plane, then identify the length of the line that the two points create.

Notice that to show point (1, 2), specify in which part of the graph does 1 and 2 meet. 1 is 1 unit to the right of 0 and 2 is 2 units up. Likewise, for point (1, -1), specify in which part of the graph does 1 and -1 meet 1 is 1 unit to the right of 0 and -1 is 1 unit down. Then, Let Q be the point (1, 2), and R be the point (1, -1).

To find the length of the line segment \(\overline{Q R}\), get the absolute value of the difference of y-coordinate of point Q  to the y-coordinate of R since points Q and R have the same x-coordinate which is 1. Then
length of \(\overline{Q R}\) = |2 – (-1)|
= |2 + 1|
= 3
Therefore, the length of \(\overline{Q R}\) is 3 units.

Question 15.
(-4, 9) and (-8, 9)
Answer:
The target of this task is to plot and connect point (1, -3) and point (5, -3) on a coordinate plane, then identify the length of the line that the two points create.

Observe that to represent point (1, -3), point out in which part of the graph does 1 and -3 meet 1 is 1 unit to the right of 0 and -3 is 3 units down. Likewise, for point (5, -3), point out in which part of the graph does 5 and -3 meet 5 is 5 units to the right of 0 and -3 is 3 units down Then, let S be the point (1, -3), and T be the point (5, -3).

Now, to find the length of the line segment \(\overline{S T}\), get the absolute value of the difference of x-coordinate 1 of point S to the x-coordinate of T since points S and T have the same y-coordinate which is -3. Then
length of \(\overline{S T}\) = |1 – 5|
= |-4|
= 4
Therefore, the length of \(\overline{S T}\) is 4 units.

Question 16.
(-2, 5) and (8, 5)
Answer:
The objective of this task is to plot and connect point (-5, 3) and point (5, 3) on a coordinate plane, then identify the
length of the line that the two points create.

Note that to show point (-5, 3), specify in which part of the graph does -5 and 3 meet -5 is 5 units to the left of 0 and 3 is 3 units up. Likewise, for point (5, 3), specify in which part of the graph does 5 and 3 meet 5 is 5 units to the right of 0 and 3 is 3 units up. Then, let U be the point (-5, 3), and V be the point (5, 3).

Next, to determine the length of the Line segment \(\overline{U V}\), get the absolute value of the difference of x-coordinate of point U to the x-coordinate of V since points U and V have the same y-coordinate 3. Then
length of \(\overline{U V}\) = |-5 – (5)|
= |-5 + -5|
= |-10|
= 10
Therefore, the length of \(\overline{U V}\) is 10 units.

Find the volume of each solid. Use 3.14 as an approximation for π. Round your answers to the nearest tenth if necessary.

Question 17.

Answer:
Volume of cone ≈ 1206.37 cm³ ,

Explanation:

Question 18.

Answer:
The volume of cylinder ≈1206.37 cm³ ,

Question 19.

Answer:
Volume of sphere is ≈33.51 cm³,

Explanation:

Question 20.

Answer:
Volume of triangular prism is 35 cm³ ,

Explanation:
Volume of triangular prism = area of cross-section X length,
5 cm²  × 7 cm = 35 cm³.

Question 21.

Answer:
Volume of Pyramid is 15 cm³,

Explanation:

Practice the problems of Math in Focus Grade 8 Workbook Answer Key Cumulative Review Chapters 5-6 to score better marks in the exam.

Math in Focus Grade 8 Course 3 A Cumulative Review Chapters 5-6 Answer Key

Concepts and Skills

Solve each system of linear equations by making tables of values. Each variable x is a positive integer less than 7. (Lesson 5.1)

Question 1.
4x + 3y = 23
y – 2x = 1
Answer:
Given the linear equations,
4x + 3y = 23
y – 2x = 1
4(y – 2x = 1)
2y – 4x = 2
4x + 3y = 23
-4x + 2y= 2
5y = 25
y = 5
substitute the value of x in the above equation.
(5) – 2x = 1
5 – 2x = 1
-2x = 1 – 5
-2x = -4
x = 2
Thus the value of x is 2 and y is 5

Question 2.
4y + 3x = 19
x + y = 5
Answer:
Given,
4y + 3x = 19
x + y = 5
Rewrite them as
3x + 4y = 19
3x + 3y = 15
–     –          –
0x + y = 4
y = 4
substitute the value of y in the above equation
x + 4 = 5
x = 5 – 4
x = 1

Solve each system of linear equations by using the elimination or substitution method. Explain your choice of method. (Lesson 5.2)

Question 3.
3a – 2b = 1
2a + 3b = 18
Answer:
Given,
3a – 2b = 1—- (×2)
2a + 3b = 18–(×3)
6a – 4b = 2
6a + 6b = 36
–    –          –
0a – 10b = -34
10b = 34
b = 3.4
substitution the value of b in the equation
3a – 2(3.4) = 1
3a – 6.8 = 1
3a = 1 + 6.8
3a = 7.8
a = 2.6
substitution method

Question 4.
0.7x – 1.2y = -11.5
0.5x + 3.5y = 31
Answer:
Given,
0.7x – 1.2y = -11.5
0.5x + 3.5y = 31
5 (0.7x – 1.2y = -11.5)
0.35x – 6y = -57.5
7(0.5x + 3.5y = 31)

0.35x + 24.5y = 217
0.35x – 6y = -57.5
–          +        +
0x + 30.5y = 274.5
30.5y = 274.5
y = 274.5/30.5
y = 9
Now substitute the value of y in the above equation.
0.5x + 3.5(9) = 31
0.5x = 31 – 31.5
0.5x = -0.5
x = -0.5/0.5
x = -1
Substitution method is used to solve the given equations.

Question 5.
\(\frac{1}{4}\)h + \(\frac{2}{3}\)k = 5
\(\frac{3}{4}\)h + k = 6
Answer:
Given,
\(\frac{1}{4}\)h + \(\frac{2}{3}\)k = 5
\(\frac{3}{4}\)h + k = 6
\(\frac{2}{3}\) \(\frac{3}{4}\)h + \(\frac{2}{3}\)k = 6 . \(\frac{2}{3}\)
Now solve both the equations
\(\frac{1}{4}\)h + \(\frac{2}{3}\)k = 5
\(\frac{1}{2}\)h + \(\frac{2}{3}\)k = 4
\(\frac{1}{4}\)h + 0 = 1
h = 1/\(\frac{1}{4}\)
h = 4
Now substitute the value of h in the above equation
\(\frac{3}{4}\)(4) + k = 6
3 + k = 6
k = 6 – 3
k = 3
The substitution method is used to solve the given equations.

Solve each system of linear equations by using the graphical method. Use 1 grid square on both axes to represent 1 unit for the interval from -5 to 10. (Lesson 5.4)

Question 6.
3x + y = 3
4x – 2y = 14
Answer:
Given,
3x + y = 3
4x – 2y = 14
3x + y = 3— × (2)
6x + 2y = 6 —- (eq. 3)
Solving eq. 3 and 2
6x + 2y = 6
4x – 2y = 14
10x = 20
x = 20/10
x = 2
3(2) + y = 3
6 + y = 3
y = 3 – 6
y = -3
3x + y = 3

X	-1	0	1
y	3(-1) + y = 3 -3 + y = 3 y = 6	3(0) + y = 3 y = 3	3(1) + y = 3 3 + y = 3 y = 0

4x – 2y = 14

X	-1	0	1
y	4(-1) – 2y = 14 -4 – 2y = 14 -2y = 18 y = -9	4(0) – 2y = 14 -2y = 14 y = -7	4(1) – 2y = 14 4 – 2y = 14 -2y = 10 y = -5

Question 7.
2x – y = 3
x + y = 9
Answer:
Given,
2x – y = 3
x + y = 9
3x = 12
x = 4
4 + y = 9
y = 9 – 4
y = 5

x	-1	0	1
y	2(-1) – y = 3 -2 – y = 3 y = 5	2(0) – y = 3 y = -3	2(1) – y = 3 2 – y = 3 y = 1

x + y = 9

x	-1	0	1
y	-1 + y = 9 y = 10	0 + y = 9 y = 9	1 + y = 9 y = 8

Question 8.
2x + y = 25
3x – 4y = – 1
Answer:
Given,
2x + y = 25— eq. 1
3x – 4y = – 1—- eq. 2
Solve the equations 1 & 2
2x + y = 25 —× 4
8x + 4y = 100—(eq. 3)
3x – 4y = – 1
Solving equations 3 and 2
8x + 4y = 100
3x – 4y = – 1
11x = 99
x = 99/11
x = 9
Substitute the value of 9 in eq. 1
2(9) + y = 25
y = 25 – 18
y = 7

Identify whether each system of linear equations is inconsistent or dependent. Justify your answer. (Lesson 5.5)

Question 9.
3x + 4y = 12
\(\frac{3}{4}\)x + y = 3
Answer:
Given,
3x + 4y = 12
\(\frac{3}{4}\)x + y = 3
3x + 4y = 12
Dependent
After multiplying equation 2 by 4, the equations are 3x + 4y = 12
Thus the equations are equivalent

Question 10.
0.2x + 1.2 = y
x + 8 = 5y
Answer:
Given,
0.2x + 1.2 = y
x + 8 = 5y
0.2x – y = 1.2 — × 5
x – 5y = 6
x – 5y = -8
0
Inconsistent

Question 11.
3x + 2y = \(\frac{1}{3}\)
9x + 6y = 1
Answer:
Given,
3x + 2y = \(\frac{1}{3}\)
9x + 6y = 1
9x + 6y = 1
Dependent
Thus the equations are equivalent

Tell whether the relation in each mapping diagram, table, or graph is a function. Explain. (Lesson 6.1)

Question 12.

Answer:
No, because the output has more than one value.

Question 13.

Answer:
Yes because each input has exactly one output, it is a function.

Question 14.

Answer:
Yes because each input has exactly one output, it is a function.

Question 15.

Answer:
No because, at least one vertical line intersects the graph at more than one point, it is not a function.

Write an algebraic equation for each function. (Lesson 6.2)

Question 16.
A chef makes 100 hot dog buns each morning. In the afternoons, he makes 40 hot dog buns per hour. The total number of hot dog buns he makes each day, y, is a function of the time he takes to make the buns in the afternoons, x hours.
Answer:
40x
Th number of hot dog buns made in the afternoons is the product between number of hours x and the rate of 40 hot dog buns per hour:

y = 100 + 40x
The total number of hot dogs buns per day is the sum between the number of hot dogs buns made in the morning and the number of hot dogs buns made in the afternoon:

Question 17.
Gina walks 2 kilometers from home to a park. She then jogs at an average speed of 7 kilometers per hour. The total distance she traveled, d kilometers, ¡s a function of the time she takes to jog, t hours.
Answer:
Given,
Gina walks 2 kilometers from home to a park. She then jogs at an average speed of 7 kilometers per hour.
The total distance she traveled, d kilometers, is a function of the time she takes to jog, t hours.
d = 2 + 7t

Question 18.

Answer:
The graph being a line, the function is Linear. Its slope-intercept equation S:
y = mx + b
We determine the slope m using the sLope formula and the points (0, 4) and (4, 16):
m = \(\frac{16-4}{4-0}\)
= \(\frac{12}{4}\)
= 3
The graph intersects the y-axis in (0, 4), so the y-intercept is 4:
b = 4
The line’s equation is:
y = 3x + 4

Question 19.

Answer: y = -10x + 40

Tell whether each table, equation, or graph represents a linear function. If so, find the rate of change. Then tell whether the function is increasing or decreasing. (Lesson 6.3)

Question 20.

Answer: Yes, It is increasing

Question 21.

Answer:
No; it is decreasing

Question 22.
3y + 2x = 10
Answer:
No, it is decreasing

Question 23.
V = l³, l > 0
Answer: no; increasing

Question 24.

Answer: Yes, it is decreasing

Question 25.

Answer:
Yes; decreasing

Problem Solving

Solve. Show your work.

Question 26.
A school librarian keeps track of the number of students in the library at any one time, and the number of computers being used. The table shows the data she collected for one day. (Chapter 6)

Draw a mapping diagram to represent the relation between the number of students in the library and the number of computers being used. Is this relation a function? Explain.
Answer:

It is a one to many relation.

Question 27.
Find two numbers whose sum is 60, such that twice the lesser number is equal to half of the greater number. (Chapter 5)
Answer:
Let the two numbers be x and 4x
x + 4x = 60
5x = 60
x = 60/5
x = 12
4x = 4(12) = 48

Question 28.
The diagram shows a parallelogram with side lengths in inches. (Chapter 5)

a) Find the values of a and b.
Answer:
3b – 4 + 2b + 3
5b – 1
b = 1/5 inch.
(3a – b) + (2a + b)
5a

b) Find the perimeter of the parallelogram.
Answer:
perimeter of the parallelogram = 2(a + b)
p = 2 (5b – 1 + 5a)
p = 10a + 10b – 2

Question 29.
Glenn donates $50 to a charity. In addition, he pledges to donate $2 per month beginning this month. The total amount he donates, y dollars, is a function of the number of months he donates, x months. (Chapter 6)
a) Write an algebraic equation for the function.
Answer: y = 50 + 2x

b) Graph the function and draw a line through the points. Use 1 unit on the horizontal axis to represent 1 month for the x interval from 0 to 6, and 1 unit on the vertical axis to represent $2 for the y interval from 50 to 62. Do the coordinates of every point on the line make sense for the function? Explain.
Answer:

Number of months	0	1	2	3	4	5	6
Total amount	50	52	54	56	58	60	62

No, only whole numbers are meaningful for the input and output. The input values, which are the number of months must be whole numbers. Hence, the corresponding output values of the function are also whole numbers.

c) Describe how the slope and y-intercept of the graph are related to the function.
Answer:
The y-intercept, 50 means that Glenn donated $50 to the charity at first. The slope 2, gives the rate that he donates per month. For every month that passes, the total amount that he donated increases by $2.

Question 30.
In a stationery store, 15 pencils and 20 rulers are sold for $20.50. Jennifer buys 7 pencils and 2 rulers for $5.90. Find the price of each item. (Chapter 5)
Answer:
Let the cost of the pencil be x
Let the cost of rulers be y
15x + 20y = 50
7x + 2y = 5.90—- × 10
70x + 20y = 59—Eq.3
Solving and eq. 3 – 1
70x + 20y = 59
15x + 20y = 50
55x = 9
x = 9/55
x = 0.16
Thus the cost of each pencil is $0.16
7(0.16) + 2y = 5.90
1.12 + 2y = 5.90
2y = 5.90 – 1.12
2y = 4.78
y = 2.39
Thus the cost of the rulers is $2.39

Question 31.
A group of 70 students and teachers visited a theme park. The teachers were charged a regular price of $50 per ticket. A 30% count of the regular ticket price was given to each student. The total ticket price for the whole group cost $2,600. Find the number of students and teachers in the group. (Chapter 5)
Answer:
Let’s note:
t = the number of teachers
s = the number of students
The total number of persons in the group:
t + s = 70   Equation 1
The cost of all tickets:
50t + (1 – 0.30)50s = 2600
50t + 35s = 2600    Equation 2
We write the system:
t + s = 70   Equation 1
50t + 35s = 2600   Equation 2
We use the substitution method
t + s = 70
We express t in terms of s from Equation 1:
t + s – s = 70 – s
t = 70 – s   Equation 3
Substitute Equation 3 in Equation 2 and determine s:
50(70 – s) + 35s = 2600
3500 – 50s + 35s = 2600
3500 – 15s = 2600
3500 – 15s – 3500 = 2600 – 3500
-15s = -900
\(\frac{-15 s}{-15}\) = \(\frac{-900}{-15}\)
s = 60
Substitute s in Equation 1 to determine t:
t = 70 – 60
= 10
The number of students is 60 and the number of teachers is 10.

Question 32.
The fuel tank in Tasha’s car can hold up to 8 gallons of gasoline. She can drive a distance of 40 miles for each galIf gasoline. The total amount of gasoline in her fuel tank, y gallons, is a function of the distance she drives, x miles. (Chapter 6)
a) Give the least possible input value and the corresponding output value. Tell whether the function is linear or nonlinear. Then tell whether the function is increasing or decreasing. Explain.
Answer:
At the initial moment the tank is full, therefore it has 8 gallons. As the car moves, the amount of gasoline decreases by 1 gallon for each 40 miles, therefore it is linear and decreasing. The Least possible input value is 0 (the initial distance). The corresponding output value is 8 gallons.
x = 0 miles
y = 8 gallons

b) Sketch a graph for the function. Identify the y-intercept of the graph.
Answer:
The graph starts at (0; 8) and falls to the right with the slope \(\frac{1}{40}\).
y = mx + b
y = –\(\frac{1}{40}\)x + 8
We graph the function.

The y-intercept is b = 8.

Question 33.
The pressure in two tanks increases at a constant rate of 2 bars per minute. The initial pressure in Tank A is 3 bars. The pressure in Tank B is 14 bars after 3 minutes. (Chapter 5)
a) Write a system of two linear equations for the pressure p in the two tanks in terms of the time t.
Answer:
The pressure in two tanks increases at a constant rate of 2 bars per minute.
The initial pressure in Tank A is 3 bars.
Tank A: p = 2t + 3
The pressure in Tank B is 14 bars after 3 minutes.
Tank B: p = 2t + 8

b) Graph the two equations on a coordinate plane. Use 1 grid square on both axes to represent 1 unit for the interval from 0 to 5.
Answer:
We graph the two functions on a coordinate plane:

c) When will the pressure in both tanks be the same? How do you know?
Answer:
The pressure in both tanks will not be the same.
The two lines are parallel.

Question 34.
Mr. Johannsen wants to compare how fast water evaporates from two inflatable swimming pools, A and B, placed at different locations. The height of the water level in each pool, h inches, is a function of the time it takes for the water to evaporate, t days. (Chapter 6)
Pool A

Pool B
An initial water level of 24 inches where water evaporates at a rate of 0.4 inch per day
a) Write an algebraic equation to represent each function.
Answer:
We are given the function for pool A:

We have:

The rate of change is:
\(\frac{-2.5}{5}\) = -0.5
\(\frac{-2.5}{5}\) = -0.5
\(\frac{-2.5}{5}\) = -0.5
\(\frac{-2.5}{5}\) = -0.5
Because the rate of change for the function is constant the table represents a linear function. Its slope is:
m = -0.5
We determine the y-intercept using the fact that the graph passes through the point (0.25):
b = 25
Substitute m, b to determine the equation:
h = -0.5t + 25
We determine the equation for Pool B, using the y-intercept 24 and the slope -0.4:
h = -0.4t + 24

b) Graph the two functions on the same coordinate grid. Use 1 unit on the horizontal axis to represent 5 days for the t interval from 0 to 20, and 1 unit on the vertical axis to represent 1 inch for the h interval from 15 to 25.
Answer:
We graph the two functions:

c) Use a verbal description to compare the two functions.
Answer:
Both functions are linear and decreasing.

The initial height of the water in Pool A is greater than that of Pool B, but the rate of water evaporation is higher for Pool A. After 10 days the height of the water in Pool A is smaller than that of Pool B.

Practice the problems of Math in Focus Grade 8 Workbook Answer Key Chapter 6 Review Test to score better marks in the exam.

Math in Focus Grade 8 Course 3 A Chapter 6 Review Test Answer Key

Concepts and Skills

Given the relation described, identify the input and the output.

Question 1.
Daphne wants to find the area of a circle given its radius.
Answer:
Input: Radius
Output: Area of Circle

Question 2.
Mr. Reynard wants to find the total cost of the number of items he bought at a store where everything costs one dollar.
Answer:
Input: one dollar
Output: total cost of the number of items

Question 3.
The head of the English department wants to see how each student in Grade 8 does on an English test.
Answer:
Input: Each student’s name
Output: Grades for the English test

Based on the mapping diagrams, state the type of relation. Tell whether each relation is a function.

Question 4.

Answer: Many to one relation, it is a function

Question 5.

Answer: One to many relation, it is not a function

Tell whether each relation is a function.

Question 6.

Answer: One to one relation, it is a function

Question 7.

Answer: Many tone relation, it is a function

Question 8.

Answer: No

Use graph paper. Show your work.

Question 9.
Represent the function y = -4x + 6 as a table and as a graph. Use 1 grid square on the horizontal axis to represent 1 unit for the x interval from -3 to 3, and 1 grid square on the vertical axis to represent 2 units for the y interval from -6 to 18.
Answer:
Given,
y = -4x + 6

x	-3	0	3
y	18	6	-6

Tell whether each function is linear or nonlinear. Then tell whether the function is increasing or decreasing.

Question 10.

Answer: Non-Linear and decreasing function

Question 11.
The area of a square, A square centimeters, is a function of its side length, s centimeters, where A = s².
Answer:
Non-linear and increasing function

Problem Solving

Describe the function. Sketch a graph for the function.

Question 12.
A large region has experienced heavy rains. Government officials decide to open a floodgate to release water from the reservoir at a constant rate of 1 cubic kilometer per hour. Before they open the gate, there are 29 cubic kilometers of water in the reservoir. The amount of water in the reservoir, y cubic kilometers, is a function of the number of hours the floodgate has been opened, x hours.
a) Give the least possible input value and the corresponding output value. Tell whether the function is linear or nonlinear. Then tell whether the function is increasing or decreasing. Explain.
Answer:
The least possible input value is the smallest value of x which is obtained at the initial moment:
x = 0
The corresponding output value y is the amount of water in the reservoir in the initial moment:
y = 29

b) Sketch a graph for the function. Identify the y-intercept of the graph.
Answer:
The graph passes through the point (0, 29), therefore the y-intercept is:
b = 29
In order to sketch a graph of the function, we use the slope m = -1 and the y-intercept 29:

Solve. Show your work.

Question 13.
The student council orders T-shirts with the school logo from an online company. The cost for each T-shirt is $2, and the shipping charge for all the shirts is $25. The student council wants to find out the total amount of money they pay, y dollars, for the number of T-shirts they order, x.
a) Write an algebraic equation to represent the function.
Answer: y = 25 + 2x

b) Use graph paper. Graph the relationship between x and y. Use 1 unit on the horizontal axis to represent 1 T-shirt for the x interval from 0 to 10, and 1 unit on the vertical axis to represent $2 for the y interval from 25 to 45.
Answer:

c) Identify whether the function is linear or nonlinear.
Answer: It is linear

d) Identify whether the function is increasing or decreasing. Explain.
Answer: It is increasing because as the values of x increase, the corresponding values of y also increase.

Question 14.
A scientist is checking to see whether pollutants are causing a decrease in oxygen levels in a river near a pipe that drains into the river. She notices that the distance downstream from the pipe, in meters, and the concentration of oxygen in the water, in milligrams per liter, can be described by the function y = 2 + 0.1x, where y is the concentration of oxygen and x is the distance from the pipe. The scientist also tested oxygen levels upstream from the pipe. The graph shows a function that represents this upstream oxygen level concentration.

a) Copy the graph shown. Then graph the function y = 2 + 0.1x on the same coordinate plane.
Answer:
We are given the downstream function:
y = 2 + 0.1x
We graph both functions:

b) Use a verbal description to compare the two functions. Give a possible reason for the difference in oxygen levels upstream and downstream from the pipe.
Answer:
The oxygen level upstream from the pipe is always at 12mg/L regardless of the distance from the pipe.
However, the oxygen level in the river downstream from the pipe is 2mg/L when the distance from the pipe is 0 m and it increases as the distance from the pipe increases. Because the concentration of oxygen increases at a rate of 0.1mg/L, the rate of change of the function is constant. So, it is an increasing linear function; One possible reason is that the river is polluted downstream from the pipe.

Practice the problems of Math in Focus Grade 8 Workbook Answer Key Chapter 6 Lesson 6.4 Comparing Two Functions to score better marks in the exam.

Math in Focus Grade 8 Course 3 A Chapter 6 Lesson 6.4 Answer Key Comparing Two Functions

Math in Focus Grade 8 Chapter 6 Lesson 6.4 Guided Practice Answer Key

Complete.

Question 1.
Water is pumped into two aquariums, P and Q. The tables show two functions relating the total amount of water, y liters, and the time taken, t minutes, to pump the water into each aquarium.

a) Use a verbal description to compare the two functions.
Aquarium P

Aquarium Q

Both functions are and functions. The function for Aquarium has a greater rate of change than the function for Aquarium .
Answer:

Both functions are linear and increasing functions. The function for Aquarium Q has a greater rate of change than the function for Aquarium P.

b) Write an algebraic equation to represent each function. Then write the initial input and output values of each function.
Aquarium P
Use the ordered pair (5, 70) and the rate of change, , to find the value of the y-intercept, b.
Linear function:

So, the algebraic equation for Aquarium P is y = .
Answer:
Let the input be t and the output be y
y = mt + b
Substitute the values for m, t and y
70 = 10(5) + b
70 = 50 + b
b = 70 – 50
b = 20

Aquarium Q
Use the ordered pair (10, 170) and the rate of change, , to find the value of the y-intercept, b.
Linear function:

So, the algebraic equation for Aquarium Q is y = .
Both functions have an initial output value of corresponding to an initial input value of 0.
Answer:
Let the input be t and the output be y
y = mt + b
Substitute the values for m, t and y
170 = 10x + 20
170 – 20 = 10x
10x = 150
x = 150/10
x = 15

c) Which of the two aquariums, P and Q, is filled with water more quickly?
Comparing the rates of change for the two shops, Aquarium has a . rate of change. This means that Aquarium will be filled with water more quickly.
Answer:
Comparing the rates of change for the two shops, Aquarium Q has a greater. rate of change. This means that Aquarium Q will be filled with water more quickly.

Complete.

Question 2.
Two classes, A and B, compare the amount of donations they will raise for a charity by participating in a walkathon. The amount of donations they will raise, y dollars, is a function of the distance the students walk, x miles.
Class A

Class B
Amount of donations: y = 20x + 50

a) Write an algebraic equation to represent the table of values representing the amount of donations Class A will raise for the charity.

The algebraic equation is y = .
Answer:

The algebraic equation is y = 15x + 100

b) Use a verbal description to compare the two functions.
Both functions are and functions. Comparing the two equations, because > 50, Class raises a greater amount of money at first.
Comparing the rates of change shows that Class A will raise $ for each mile the students walk, and Class B will raise $ for each mile the students walk. So, the amount of donations Class will raise increases more quickly than the amount of donations Class will raise.
Answer:
Both functions are linear and increasing functions. Comparing the two equations, because 100 > 50, Class A raises a greater amount of money at first.
Comparing the rates of change shows that Class A will raise $ 15 for each mile the students walk, and Class B will raise $ 20 for each mile the students walk. So, the amount of donations Class B will raise increases more quickly than the amount of donations Class A will raise.

Math in Focus Course 3A Practice 6.4 Answer Key

Tell whether the equation y = -2x + 3 can represent each of the following functions.

Question 1.

Answer: Yes
y = -2x + 3
y = -2(2) + 3
y = -4 + 3
y = -1
y = -2(3) + 3
y = -6 + 3
y = -3
y = -2x + 3
y = -2(-1) + 3
y = 2 + 3
y = 5

Question 2.

Answer: No

Question 3.

Answer: No

Tell whether each function can represent the table of values.

Question 4.
y = 3x – 4
Answer: No

Question 5.
y = 2x – 5
Answer: No

Question 6.

Answer: yes

Tell whether each function represents the verbal description.

Bryan has $30 in savings at first. He wants to save $5 per month beginning this month. y represents his total savings, in dollars, and x represents the number of months he saves.
Question 7.

Answer: No

Question 8.
y = 30 + 5x
Answer: Yes

Question 9.
y = 30 – 5x
Answer: No

Solve. Show your work.

Question 10.
Clara and Elaine have some savings. The functions that relate each girl’s total savings, y dollars, to the number of months, x, that each girl saves are as follows:
Clara: y = 380 + 20x
Elaine: y = 400 + 15x
a) Use a verbal description to compare the two functions.
Answer:
Clara has 380 dollars in her savings. She wants to save 20$ from this month.
Elaine has 400 dollars in her savings. She wants to save 15$ from this month.

b) Graph the two functions on the same coordinate plane. Use 1 unit on the horizontal axis to represent 1 month for the x interval from 0 to 8, and 1 unit on the vertical axis to represent $20 for the y interval from 380 to 540. For each function, draw a line through the points.
Answer:
We graph both functions:

c) Who will save more over time? Explain.
Answer: Clara saves more money than Elaine.

Question 11.
The director of a theater group wants to rent a theater for an upcoming show. The director has two options for paying for the rental. Both options involve paying a deposit and then paying an additional charge for each ticket sold. For each function, the total amount the director would pay, y dollars, is a function of the number of tickets sold, x.
Option A

Option B
A deposit of $800 plus $6 per ticket sold.
a) Write an algebraic equation to represent each function.
Answer:
Option A: y = 1000 + 4x
Option B: y = 800 + 6x

b) Use a verbal description to compare the two functions.
Answer:
Both functions are a linear and increasing functions. Comparing the two equations, because 1000 > 800, Option A costs more at first. Comparing the rates of change shows that the total fee for option A increases by $4 for each ticket sold, and the total fee for option B increases by $6 for each ticket sold, and the total fee for the director will pay for Option B will increase more quickly than the total fee for Option A as the number of tickets sold increases.

c) Math Journal The theater seats up to 200 people. If the director expects to sell all the tickets, which of the two options, A or B, offers a better deal? Explain.
Answer:
The total fee for option A is lower than the total fee for Option B when all the tickets are sold.

Question 12.
A factory needs to grate at least 8,000 pounds of cheese each day.

The manager of the factory needs to buy a new cheese grating machine. She is trying to decide between Machine A and Machine B. The functions shown describe how many pounds of cheese, y, are left t minutes after each machine starts grating an initial batch of cheese.
Machine A
The function is y = 2,000 – 80t. The initial value of 2,000 pounds represents the weight of each batch of cheese to be grated. After one batch is grated, another batch can be added to the machine.

Machine B

a) Write an algebraic equation to represent the function for Machine B.
Answer:
The graph of the function for Machine B passes through the points (0, 8000) and (80, 0) We determine the slope of the line using the slope formula:
m = \(\frac{0-8000}{80-0}\)
= \(\frac{-8000}{80}\)
= -100
The graph of the function for Machine B passes through the point (0, 8000), therefore the y-intercept is:
b = 8000
The equation for Machine B is:
y = -100t + 8000

b) Math Journal Assuming that the machines are of the same quality, which machine would you recommend that the manager buy? Explain.
Answer:
The equation for Machine A is:
y = -80t + 2000
We graph the two functions:

We notice that with the Machine from Option A the manager would grate 4 • 2000 = 8000 pounds of cheese in 100 minutes, while with the Machine from Option B he can grate 8000 pounds in only 80 minutes, so Option B is better.

Brain @ Work

Five teachers at a school brought a group of students to a museum exhibit.

For group tours involving any number of adults and at least 5 students, the museum offers three packages, A, B, and C. The functions shown below represent the total admission fee, y dollars, that five teachers and x students will pay to see the exhibit.
Package A
Each adult ticket costs $30 and each student ticket costs $15.

Package B

Package C
Each adult ticket costs $60 and each student ticket costs $12.

Question 1.
Use an algebraic equation to represent each of the three functions.
Answer:
Equation for Package A: y = 150 + 15x
Package B: y = 250 + 14x
Package C: y = 300 + 12x

Question 2.
Graph the three functions on the same coordinate plane. Use 1 unit on the horizontal axis to represent 5 students for the x interval from 0 to 50, and 1 unit on the vertical axis to represent $50 for the y interval from 150 to 950. For each function, draw a line through the points.
Answer:

Practice the problems of Math in Focus Grade 8 Workbook Answer Key Chapter 6 Lesson 6.3 Understanding Linear and Nonlinear Functions to score better marks in the exam.

Math in Focus Grade 8 Course 3 A Chapter 6 Lesson 6.3 Answer Key Understanding Linear and Nonlinear Functions

Math in Focus Grade 8 Chapter 6 Lesson 6.3 Guided Practice Answer Key

Tell whether each table of values represents a linear or nonlinear function. Explain.

Question 1.

Because the rate of change for the function is , the table represents a function.
Answer:

Because the rate of change for the function is constant, the table represents a linear function.

Question 2.

Because the rate of change for the function is , the table represents a function.
Answer:
Because the rate of change for the function is non-constant, the table represents a non-linear function.

Tell whether each graph represents a linear function. If so, find the rate of change.

Question 3.

Because the graph is a , it represents a function.
The line passes through (, ) and (, ).
Rate of change = \(\frac{?-?}{?-?}\)
=
So, the rate of change of the graph is .
Answer:
Because the graph is a straight line, it represents a linear function.
The line passes through (0, 3) and (6, 8).
Rate of change = \(\frac{8-3}{6-0}\)
= 5/6
So, the rate of change of the graph is constant.

Question 4.

Because the graph is a , it represents a function.
Answer:
Because the graph is a curve, it represents a non-linear function.

Describe the function. Sketch a graph for the function.

Question 5.
A cruise ship traveling at a constant speed consumes 4,000 gallons of gasoline per hour. When fully filled, it has a fuel capacity of 330,000 gallons. The amount of gasoline consumed, y gallons, is a function of the total traveling time, x hours.
a) Give the least possible input value and the corresponding output value. Tell whether the function is linear or nonlinear. Then tell whether the function is increasing or decreasing. Explain.
Answer:
At the beginning of the trip, the cruise ship has 330,000 gallons of gasoline.
So, the least possible input value is 0 and the corresponding output value is 330,000.
For every hour of traveling time, the cruise ship consumes 4,000 gallons of gasoline.
So, the rate of change of the function is constant.
As the total traveling time increases, the amount of gasoline left decreases.
Thus the function is a linear and decreasing function.

b) Sketch a graph for the function.
Answer:

Hands-On Activity

Materials:
number cards (from -5 to 5)

SKETCH LINEAR FUNCTIONS

Work in pairs.

STEP 1: Shuffle the cards and place them face down on the table.

STEP 2: Each player draws two cards. Use your cards to write an equation in slope-intercept form. Use one of the cards for the slope and one for the y-intercept of the equation.
For example:

Slope-intercept form: y = -2x + 3

STEP 3: Graph the equation you wrote.

STEP 4: Copy and complete the table. For each equation that you and your partner write, record the slope, the y-intercept, and whether the function is increasing or decreasing.

STEP 5: Repeat STEP 2 to STEP 4 until a player has written two equations that represent increasing functions and two equations that represent decreasing functions. If no player has done this, reshuffle the cards and repeat STEP 2 to STEP 4. The player who reaches this goal first wins the game.

Math Journal What can you conclude about the slope of the graph of an increasing function and the slope of the graph of a decreasing function? Explain.
Answer:

Math in Focus Course 3A Practice 6.3 Answer Key

Tell whether each table of values represents a linear or nonlinear function.

Question 1.

Answer: Linear function

Question 2.

Answer:

Linear Function

Question 3.

Answer:

Non-linear function

Question 4.

Answer:

Tell whether each graph represents a linear function. If so, find the rate of change.

Question 5.

Answer:
Yes, 1/2

Question 6.

Answer: Non-linear and decreasing

Tell whether each function is linear or nonlinear. Then tell whether the function is increasing or decreasing.

Question 7.

Answer:
Non-linear and increasing function

Question 8.

Answer: straight line and linear function

Question 9.

Answer: Non-linear and decreasing function

Question 10.

Answer: Straight line and linear function

Describe the function. Sketch a graph for the function.

Question 11.
A machine at a factory pours juice into bottles at a constant rate of 6 liters per minute. The total amount of juice poured, y liters is a function of the number of minutes that the juice is poured, x.
a) Give the least possible input value and the corresponding output value. Tell whether the function is linear or nonlinear. Then tell whether the function is increasing or decreasing. Explain.
Answer:
If the time taken for the machine to pour juice into bottles is 0 min, the amount of juice poured will be 0L.
So, the least possible input value is 0 and the corresponding output value is 0.
Because the machine pours juice at 6L/min, the rate of change of the function is constant. As the time taken for the machine to pour juice into bottles increases, the amount of juice poured also increases.
Hence the function is a linear and increasing function.

b) Sketch a graph for the function.
Answer:

Question 12.
Aidan was 100 miles from Town P. He traveled to Town P by car at a constant speed. The distance from Town P, y miles, is a function of the traveling time, x hours.
a) Give the least possible input value and the corresponding output value. Tell whether the function is linear or nonlinear. Then tell whether the function is increasing or decreasing. Explain.
Answer:
When Aidan [eaves to Town P, the corresponding distance from his position to Town P is 100, therefore the least possible input value is 0 and the corresponding output is 100.

Because each hour he is driving by the same speed, the rate of change of the function is constant.

As time passes (the input increases), the distance to Town P (the output) decreases, so the function is decreasing.
Therefore the function is linear and decreasing.

b) Sketch a graph for the function.
Answer:
The distance to Town P starts at (0, 100) and falls from left to right The slope of the line, which is constant, is the distance traveLed each hour (the speed):

Solve. Use graph paper.

Question 13.
The table shows the number of students, y, as a function of the number of teachers, x.

a) Tell whether the function is linear or nonlinear. Then tell whether the function is increasing or decreasing. Explain.
Answer:
The function has a constant rate of change, 25.
Thus it is a linear and increasing function.
As the number of students increases, the number of teachers also increases.

b) Graph the table of values and draw a line through the points. Use 1 unit on the horizontal axis to represent 1 teacher for the x interval, and 1 unit on the vertical axis to represent 25 students for the y interval from 100 to 250. Do the coordinates of every point on the line make sense for the function? Explain.
Answer:

No, because the number of teachers and students must be whole numbers.

Question 14.
A cyclist starts riding from home to another town. His cycling speed, y miles per hour, is a function of the amount of time he takes to cycle, x hours.

a) Tell whether the function is linear or nonlinear. Then tell whether the function is increasing or decreasing. Explain.
Answer:
Non-linear function and decreasing

b) Graph the table of values and draw a curve through the points. Use 1 unit on the horizontal axis to represent 1 hour for the x interval, and 1 unit on the vertical axis to represent 1 mile per hour for the y interval. Do the coordinates of every point on the curve make sense for the function? Explain.
Answer:

Practice the problems of Math in Focus Grade 8 Workbook Answer Key Chapter 6 Lesson 6.2 Representing Functions to score better marks in the exam.

Math in Focus Grade 8 Course 3 A Chapter 6 Lesson 6.2 Answer Key Representing Functions

Math in Focus Grade 8 Chapter 6 Lesson 6.2 Guided Practice Answer Key

Complete

Question 1.
A game shop rents out video games at a rate of $6 per game. The total amount of money the shop collects, y dollars, is a function of the number of games, x, that the shop rents out.
a) Write a verbal description of the function. Then write an algebraic equation for the function.
Total amount of money the shop collects equals the product of the rental rate and the .
= 6 • Write an equation.
= Simplify.
Answer:
Given,
A game shop rents out video games at a rate of $6 per game.
The total amount of money the shop collects, y dollars, is a function of the number of games, x, that the shop rents out.
y = 6 • x
y = 6x

b) Construct a table of x- and y-values for the function. this situation,

Answer:

c) Use the table of values in b) to graph the function. Use 1 unit on the horizontal axis to represent 1 game for the x interval, and 1 unit on the vertical axis to represent $6 for the y interval.
Answer:

Question 2.
A fire sprinkler sprays water at a rate of 8 gallons per minute. The total amount of water being sprayed, y gallons, is a function of the number of minutes, x, that the sprinkler sprays water.
a) Write a verbal description of the function. Then write an algebraic equation for the function.
Total amount of water being sprayed equals product of the rate of water flow and the .
= • Write an equation.
= Simplify.
Answer:
Given,
A fire sprinkler sprays water at a rate of 8 gallons per minute.
The total amount of water being sprayed, y gallons, is a function of the number of minutes, x, that the sprinkler sprays water.
y = 8 • x
y = 8x

b) Construct a table of x- and y-values for the function.

Answer:

c) Use the table of values in b) to graph the function. Use 1 unit on the horizontal axis to represent 1 minute for the x interval, and 1 unit on the vertical axis to represent 8 gallons for the y interval.
Answer:

Question 3.
The table shows the total distance, y miles, indicated on the odometer of Jason’s car and the amount of gasoline used, x gallons, on a particular day.

a) Graph the function. Use 1 unit on the horizontal axis to represent 1 gallon for the x interval from 0 to 5, and 1 unit on the vertical axis to represent 30 miles for the y interval from 1,000 to 1,150.
Answer:

b) Write an algebraic equation for the function.
Answer: y = 30x + 1000

c) Describe how the slope and the y-intercept of the graph are related to the function.
Answer:
The y-intercept, 1000 means that the odometer of Jason’s car shows 1000 mi when he starts driving.
The shape 30, gives the rate at which the distance indicated on the odometer is changing. For every gallon of gasoline, the distance indicated on the odometer increases by 30 mi.

Math in Focus Course 3A Practice 6.2 Answer Key

Write a verbal description of each function. Then write an algebraic equation for the function.

Question 1.
Gordon is traveling at a constant speed of 80 kilometers per hour. The distance he travels, d kilometers, is a function of the amount of time he takes to travel, t hours.
Answer:
The distance Gordon travels equals 80km/h times the number of hours he takes to travel,
d = 80t

Question 2.
Mr. Henderson pays a monthly charge of $40 for a family cell phone plan. Each additional family member pays $10 every month. The total amount Mr. Henderson and his family members pay each month, y dollars, is a function of the number of the additional family members who use the plan, x.
Answer:
Given,
Mr. Henderson pays a monthly charge of $40 for a family cell phone plan.
Each additional family member pays $10 every month.
The total amount Mr. Henderson and his family members pay each month, y dollars, is a function of the number of the additional family members who use the plan, x.
y = 10x + 40

Question 3.
Math Journal In questions 1 and 2 tell whether all values for the input and output are meaningful for the functions. Explain.
Answer:
All the values for the input and output are meaningful because time and distance are continuous quantities.
Only whole numbers are meaningful for the input and output.
The input values which are the numbers of additional family members must be whole numbers.
So, the corresponding output values of the function

Write an algebraic equation for each function. Then construct a table of x- and y-values for the function.

Question 4.
The students from the Robotics Club are making model windmills for a workshop. Each windmill has three blades. The total number of blades needed, y, is a function of the number of windmills they make, x.
Answer:
Given,
Each windmill has three blades.
The total number of blades needed, y, is a function of the number of windmills they make, x.
y = 3x

Question 5.
A newly made glass vase has a temperature of 580°C. Its temperature then decreases at an average rate of 56°C per minute. The temperature of the glass vase, y°C, is a function of the number of minutes its temperature has been decreasing, x.
Answer: y = 580 – 56x

X	-1	2	3
y	524	468	412

Each of the following graphs represents a function. Write an algebraic equation to represent the function.

Question 6.

Answer: y = 3

Question 7.

Answer: y -5/2 x + 20

Use the table of values to plot a graph to represent the function.

Question 8.
The table shows the number of chairs in a classroom, y, as a function of the number of students in the classroom. Use 1 unit on the horizontal axis to represent 1 student for the x interval, and 1 unit on the vertical axis to represent 4 chairs for the y interval.

Answer:

Use the table of values to plot a graph to represent the function. Then write an algebraic equation for the function.

Question 9.
A motorcyclist rode at a constant speed from City A to City B, which are 240 miles apart. The table shows his distance from City B, y miles, as a function of the number of hours he rode, x hours. Use 1 unit on the horizontal axis to represent 1 hour for the x interval, and 1 unit on the vertical axis to represent 40 miles for the y interval.

Answer:

y = -40x + 240

Solve. Show your work.

Question 10.
The graph shows the temperature of a package of food, y°C, as a function of the time the food is in the freezer, x minutes.

a) Write an equation in slope-intercept form to represent the function.
Answer:
The line passes through the points (0, 30) and (2, 20).
Use the slope formula:
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Substitute values:
= \(\frac{20-30}{2-0}\)
Subtract:
= \(\frac{-10}{2}\)
Simplify:
= -5
The line intersects the y-axis in point (0, 30). So the y-intercept is 30.
b = 30
Substitute the values of m and b. The slope-intercept form is:
y = -5x + 30
So an equation of the Line is y = -5x + 30.

b) What information do the values for slope and y-intercept give you about the function?
Answer:
The y-intercept shows that the initial temperature of the package was 30° C. The slope, -5, gives the rate at which the package’s temperature decreases each minute.

Question 11.
Hillary has $60 on her bus card. Every time she rides a bus, $1.50 is deducted from the value on her card. The amount of money she has on her card, y dollars, is a function of the number of times she rides a bus, x.
a) Write a verbal description of the function. Then write an algebraic equation for the function.
Answer:
The amount of money Hillary has on her bus card equals $60 minus $1.50 times the number if times she rides on a bus;
y = 60 – 1.5x

b) Construct a table of x- and y-values for the function in a). Use values of x from 0 to 6.
Answer:

Number of bus rides	0	1	2	3	4	5	6
Amount of money on Hillary’s Card (y)	60	58.5	57	55.5	54	52.5	51

c) Use the table of values in b) to plot a graph to represent the function. Use 1 unit on the horizontal axis to represent 1 bus ride for the x interval from 0 to 6, and 2 units on the vertical axis to represent $3 for the y interval from 51 to 60.
Answer:

d) How many bus rides has Hillary taken if she has $51 left on her card?
Answer: 6

Practice the problems of Math in Focus Grade 8 Workbook Answer Key Chapter 6 Lesson 6.1 Understanding Relations and Functions to score better marks in the exam.

Math in Focus Grade 8 Course 3 A Chapter 6 Lesson 6.1 Answer Key Understanding Relations and Functions

Math in Focus Grade 8 Chapter 6 Lesson 6.1 Guided Practice Answer Key

Complete.

Question 1.
Describe the relation between the inputs and the outputs.

The relation between the inputs and the outputs is a -to- relation.
Answer:
The relation between the inputs and the outputs is a many to one relation.
Because all the inputs are connected to the single output.

Question 2.
Describe the relation between the inputs and the outputs.

The relation between the inputs and the outputs is a -to- relation.
Answer:
The relation between the inputs and the outputs is a many to many relation.

Question 3.
The table shows the relation between the heights of five statues and their weights.

Copy and complete the mapping diagram to show the relation between the heights of the five statues and their weights. Then identify the type of relation between the heights and the weights.

The relation between the heights and the weights is a -to- relation.
Answer:
The relation between the heights and the weights is a one -to- one relation.

Tell whether each relation is a function. Explain.

Question 4.

Because the mapping diagram shows a -to- relation, it a function.
Answer:
Because the mapping diagram shows a one to many relation, it is a one to many function.

Question 5.

Because the mapping diagram shows a -to- relation, it a function.
Answer:
Because the mapping diagram shows a one to many relation, it is a one to many function.

Question 6.

Because the table shows a -to- relation, it a function.
Answer:
Because the table shows a one to one relation, it is a one to one function.

Tell whether the relation represented by each graph is a function. Explain.

Question 7.

Answer:
You can graph relation by writing and graphing orders pairs.
ordered pairs (1, 2), (2, 2), (3, 5), (4, 2)

From the above graph, we can say that it is a many to one relation and it is a many to one function.
The above graph is not a function.

Question 8.

Answer:
The ordered pairs are (1,5), (2,4), (3,1), (3,3), (4,2)

It is a many to one relation.
By seeing the above graph we can say that the relation is not a function.

Question 9.

Answer: The above graph is a function.

Question 10.

Answer: The above graph is not a function.

Math in Focus Course 3A Practice 6.1 Answer Key

Given the relation described, identify the input and the output.

Question 1.
Mrs. Thomas wants to find out the price charged for the same stereo speaker at different stores.
Answer:
Given
Mrs. Thomas wants to find out the price charged for the same stereo speaker at different stores.
the price of the stereo speaker is input and stores is the output.

Question 2.
Five students, Jessie, Patrick, Wayne, Colin, and Susie, have different heights. Their teacher wants to know their heights.
Answer:
the number of students is input and heights is the output.

Question 3.
Ginny wants to know what after-school activities each of her friends signed up for so she knows whether she shares the same interests.
Answer:
Ginny wants to know what after-school activities each of her friends signed up for so she knows whether she shares the same interests.
Friends is input and after-school activities is output.

Based on the mapping diagram, state the type of relation.

Question 4.

Answer:
It is a many to many relation.

Question 5.

Answer:
It is a one to many relation.

Question 6.

Answer:
It is a one to one relation.

Draw a mapping diagram to represent each relation. Then identify each type of relation.

Question 7.
The table shows the numbers of various types of fruit sold in a supermarket. Draw a mapping diagram to represent the relation between each fruit and the number sold by the supermarket. Identify the type of relation between the fruit and the number sold.

Answer:

Its is Many to one relation.

Question 8.
The table shows the scores of a soccer team playing in eight different games. Each game is represented by a number.

Draw a mapping diagram to represent the relation between the score for each game and the game number. Identify the type of relation between the score and the game number.
Answer:

One to many relation

Tell whether each statement is True or False. Explain.

Question 9.
A function is a type of relation.
Answer: true

Explanation: A function is a correspondence between two sets called domain and range.

Question 10.
All relations are functions.
Answer: False

Question 11.
Only a many-to-one relation is a function.
Answer: False

Question 12.
A one-to-many relation is a function.
Answer: True

Identify the type of relation in each mapping diagram. Then tell whether the relation is a function. Explain.

Question 13.

Answer: It is one to one relation. Thus the relation is a one to one function.

Question 14.

Answer: It is one to many relation. The relation is not a function.

Question 15.

Answer: It is a many to one relation.

Tell whether the relation represented by each graph is a function. Explain.

Question 16.

Answer: It is a many to one relation and it is not a function.

Question 17.

Answer: It is a one to many relation, it is not a function.

Question 18.

Answer: The above graph is a many to function.

Question 19.

Answer: The above graph is a many to function.

Tell whether the relation described is a function. Use a graph to support your answer.

Question 20.
The cost, y dollars, of some cheese that costs $3 per pound varies directly with the weight, x pounds, of the cheese. Use 1 unit on the horizontal axis to represent 1 pound for the x interval from 0 to 6, and 1 unit on the vertical axis to represent $3 for the y interval from 0 to 18.
Answer:

We build a table of values

We draw the graph

We apply the vertical test: as any vertical line intersects the graph in exactly one point, the relation is a function
Function

Question 21.
A tank contains 3 liters of water. The water in the tank is draining out at a rate of 0.5 liter per hour. Use 1 unit on the horizontal axis to represent 1 hour for the x interval from 0 to 6, and 1 unit on the vertical axis to represent 0.5 liter for the y interval from 0 to 3.0.
Answer:

We build a table of values

We draw the graph

We apply the vertical test: as any vertical line intersects the graph in exactly one point, the relation is a function.
Function

Question 22.
A veterinarian weighed some puppies to see if weight depends on age. The table shows the ages, x months, and the weights, y pounds, of the puppies. Use 1 unit on the horizontal axis to represent 1 month for the x interval, and 1 unit on the vertical axis to represent 1 pound for the y interval.

Answer:

we are given the table of values

we draw the graph

We apply the vertical test: as there is a vertical line which intersects the graph in more than one point, the relation is not a function.
Not a function

Solve. Show your work.

Question 23.
The table shows the number of computers the students have and the number of students in eight schools.

a) Draw a mapping diagram to represent the relation between the number of computers and the number of students.
Answer:
We draw a mapping diagram to represent the relation between the number of computers and the number of students:

b) From the mapping diagram, identify the relation between the number of computers and the number of students.
Answer: One to many relation

c) Tell whether the relation represented by the mapping diagram is a function. Explain.
Answer: No, because two inputs have more than one output each, it is not a function.

Question 24.
Math Journal Is the relationship between the side length of a square and the area of the square an example of a function? Explain.
Answer:
The relation between the side length x of a square and the area of the square x² is a one-to-one relation because to each value of x corresponds only one value of x².
Therefore the relation is a function.
Function

The table below shows the number of books sold by each of six bookstores and the sales made by each store in a week. Use the table to answer questions 25 to 27.

Question 25.
Draw a mapping diagram to represent a relation between the bookstores and the number of books they sold in the week. Identify the type of relation between the bookstores and the number of books sold. Then tell whether the relation represented by the mapping diagram is a function. Explain.
Answer:
Many to one relation, it is a function, each input has only one output.

Question 26.
Draw a mapping diagram to represent the relation between the sales made by the bookstores and the number of books sold in the week. Identify the type of relation between the sales made by the bookstores and the number of books sold. Then tell whether the relation represented by the mapping diagram is a function. Explain.
Answer:
We draw a mapping diagram to represent a relation between the sales made by bookstores and the number of books sold each week:

From the diagram we notice that the relation is one-to-many. Therefore the relation is not a function.
Not a function

Question 27.
Math Journal The store owners want to know if the relation between the number of books sold and the sales made by the bookstores is a function.
a) Draw a mapping diagram, with the number of books sold as the input, and the sales made by each bookstore as the output. Is the relation a function? Explain.
Answer:
No, One input, 523, more than one output, 2,317 and 2,569.

b) Why might one bookstore get more money for selling the same number of books as another bookstore?
Answer:
One bookstore could be selling books that are more expensive.