Math in Focus

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

4x – 2y = 14

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 + y = 9

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:

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

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

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:

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.

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

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

Math in Focus Grade 8 Chapter 6 Quick Check Answer Key

Write an algebraic expression for each of the following.

Question 1.
Benedict has 7 packs of cards. Each pack has x cards. He gives 3 cards to his sister. Write an algebraic expression for the number of cards that he has left.
Answer:
Given,
Benedict has 7 packs of cards.
Each pack has x cards.
He gives 3 cards to his sister.
7x – 3
Thus the algebraic expression for the number of cards that he has left is 7x – 3.

Question 2.
y highlighters are shared equally among 9 students. One of the students, Jessie, then buys another 3 highlighters. Write an algebraic expression for the number of highlighters she has in total.
Answer:
Given,
y highlighters are shared equally among 9 students.
One of the students, Jessie, then buys another 3 highlighters.
y/9 + 3

Evaluate each expression for the given value of the variable.

Question 3.
5x + 7 when x = -3
Answer:
Given the expression 5x + 7
when x = -3
5(-3) + 7
= -15 + 7
= -8

Question 4.
-4x – 1 when x = 3
Answer:
Given the expression -4x – 1
when x = 3
-4(3) – 1
-12 – 1 = -13

Question 5.
3 – \(\frac{1}{2}\)x when x = -5
Answer:
Given the expression 3 – \(\frac{1}{2}\)x
when x = -5
3 – \(\frac{1}{2}\)(-5)
3 + \(\frac{5}{2}\)

Question 6.
\(\frac{3}{4}\)x – 2 when x = 7
Answer:
Given the expression \(\frac{3}{4}\)x – 2
when x = 7
\(\frac{3}{4}\)(7) – 2
\(\frac{21}{4}\) – 2

This handy Math in Focus Grade 8 Workbook Answer Key Chapter 10 Review Test detailed solutions for the textbook questions.

Math in Focus Grade 8 Course 3 B Chapter 10 Review Test Answer Key

Concepts and Skills

Draw scatter plot for each of the given table of bivariate data.

Question 1.
Use 2 centimeters on the horizontal axis to represent 1 unit. Use 1 centimeter on the vertical axis to represent 5 units for the y interval from 50 to 110.

Answer:

Question 2.
Use 2 centimeters on the horizontal axis to represent 10 units for the x interval from 90 to 120. Use 1 centimeter on the vertical axis to represent 5 units for the y interval from 60 to 100.

Answer:

Describe the association between the bivariate data shown in each scatter plot.

Question 3.

Answer:
The association between the bivariate data shown in the above scatter plot is strong, linear and negative.

Question 4.

Answer: The association between the bivariate data shown in the above scatter plot is strong, non-linear, and negative.

State the line that represents the line of best fit for each scatter plot.

Question 5.

Answer: The best line fit from the above scatter plot is line B.

Question 6.

Answer: The best line fit from the above scatter plot is line B.

Identify the outlier(s) in each scatter plot.

Question 7.

Answer: (4, 2) is the outlier.

Question 8.

Answer: (4, 1) is the outlier

Construct each scatter plot and draw a line of best fit for the given table of bivariate data.

Question 9.
Use 1 centimeter on the horizontal axis to represent a score of 10. Use 1 centimeter on the vertical axis to represent a score of 5.

Answer:

Question 10.
Use 1 centimeter on the horizontal axis to represent 1,000 products. Use 1 centimeter on the vertical axis to represent $50.

Answer:

Question 11.
Use 2 centimeters on the horizontal axis to represent 5 inches. Use 1 centimeter on the vertical axis to represent 5 inches for the y interval from 35 to 90.

Answer:

Identify whether the given data is categorical or quantitative.

Question 12.
Brown, green, blue
Answer: categorical

Question 13.
$1, $2, $3, $4
Answer: quantitative

Question 13.
1 A.M., 2 A.M., 3 A.M.
Answer: categorical

Use the two-way table to answer questions 15 to 18.

Question 15.
Copy and complete the two-way table.
Answer:

Question 16.
Describe the association between the two categorical data.
Answer: The association between the two categorial data describes number of people like and do not like jogging and the number of people like swimming and do not like swimming.

Question 17.
Find the relative frequencies among the rows, and interpret their meanings. Round your answer to the nearest hundredth where necessary.
Answer:
Jogging:
156/196 = 0.80
40/196 = 0.20
72/204 = 0.35
132/204 = 0.65

Question 18.
Find the relative frequencies among the columns, and interpret their meanings. Round your answer to the nearest hundredth where necessary.
Answer:
156/228 = 0.68
72/228 = 0.32
40/172 = 0.23
132/172 = 0.77

Question 19.
Construct a two-way table using the data below.

M represents participarted in a marathon
NM represents not participarted in a marathon
F represents member of a fitness club
NF represents non-member of a fitness club
Answer:

Problem Solving

Refer to the scenario below to answer questions 20 to 27.

A bank wants to reduce the number of hours that its tellers work per month. To do this, more Automated Teller Machines (ATMs) are installed in the branch offices. The table below shows the number of ATMs, x, in the branch offices and the corresponding number of hours per month, y, that its tellers work.

Question 20.
Use graph paper to construct the scatter plot for the above bivariate data. Use 1 centimeter on the horizontal axis to represent 1 ATM. Use 1 centimeter on the vertical axis to represent 10 hours for the y interval from 80 to 230.
Answer:

Question 21.
Describe the association between the number of hours that tellers work and the number of ATMs.
Answer: By seeing the above graph we can say that as the number of ATMs increases, the number of hours that tellers work decreases.

Question 22.
Identify the outlier(s).
Answer:(10,110)

Question 23.
Draw a line of best fit.
Answer:

Question 24.
Write an equation for the line of best fit.
Answer:
The target of this task is to give the equation of the Line of best fit for the scatter pLot of the number of ATMs to the number of hours teller works.

For the equation of the line of best fit, use points that the best fit Une passes through, (7, 156) and (12, 88), to solve for the slope of the line. The slope of the line equates the difference of y₂ to y₁ over the difference of x₂ to x₁. Then, let m be the slope, (7, 156) be (x₁, y₁) and(12. 88) be (x₂, y₂) so
m = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)   Slope Formula
= \(\frac{88-156}{12-7}\)  Subtraction
= \(\frac{-68}{5}\)  Subtract
= -13.6   Simplify
Next, use the slope-intercept form to determine the equation of the straight Line given by y = mx ± b, where m is the slope, b is the y-intercept, and (7, 156) be (x, y) then
y = mx + b     Slope-intercept Form.
156 = -13.6(7) + b     Substitution.
156 = -95.2 + b     Multiply.
156 + 95.2 = -95.2 + b + 95.2    Add 95.2 to both sides.
251.2 = b    Simplify.
Lastly, using the slope-intercept form again, [et rn be the sLope equals to -13.6 and b equals to 251.2. thus
y = mx + b   Slope-intercept Form.
y = -13.6x + 251.2 Substitution.
Therefore, the equation of the line of best fit is y = -13.6x + 251.2.

Question 25.
Using the equation in 24, predict the number of teller hours required per month when there are 2 ATMs.
Answer:
The goal of this task is to give the approximate hours the tellers wiLl work per month using the graph if there are 2 Automated Teller Machines.

To estimate the hours tellers work y given the number of ATMs x, study the graph then note at what point does x equals to 2 meets y. The point is on (2, 224). Therefore, if there only exist 2 Automated Teller Machines in the branch office, then telLers would have to work around 224 hours per month.

Question 26.
Using the equation in 24, predict the number of teller hours required per month when there are 15 ATMs.
Answer:
The objective of this task is to calculated using the equation of the Line y = -13.6x + 251.2 the total hours the tellers will work per month if there are 15 Automated Teller Machines.

To compute the tellers hours per month if there exist 15 ATMs in the office branch, note that the teller hours is denoted as y and the number of ATMs is denoted as x. Then when x is 15.
y = -13.6x + 251.2    Equation of the line.
= -13.6(15) + 251.2    Substitution.
= -204 + 251.2    Multiply.
= 47.2    Simplify.
Therefore, when using the equation y = -13.6z + 251.2. the total hours the tellers work would be 47.2 if there are 15 ATMs.

Question 27.
Math Journal Explain why the equation is 24 cannot be used to predict the number of teller hours required per month for more than 30 ATMs. Discuss the accuracy of the prediction.
Answer:
The goal of this task ¡sto prove that the equation y = -13.6x + 251.2 to make a prediction for the length of working hours when there are 20 Automated Teller Machines, then describe how accurate this kind of prediction is. If 20 is to be substituted to the equation,
y = -13.6(20) + 251.2 = -20.8
The negative value of time that the equation provided makes it impossible to utilize the equation. Next rote that the association of number of ATMs to the number of hours teller works is weak, negative and linear The weak association makes it hard to tell a relationship or pattern of the two variables. Therefore, the prediction is not very right.

Refer to the scenario below to answer questions 28 to 32.

A survey is conducted to find out if providing nutrition information on the menu affects whether patrons recommend the restaurant to others.

P represents provide nutritional information
NP represents do not provide nutritional information
R represents recommend
NR represents do not recommend

Question 28.
Construct a two-way table using the above data.
Answer:
The objective of this task ¡s to make a two-way table of the survey about being aware of the nutrition of the foods and the customer recommendation.

Since there are two categories in the survey, the information about the nutrients of the food and the customer recommendation, then the two-way table will. be divided into these categories. Observe that there are 6 who are provided with information and recommended the restaurant, S was not given an information about the nutrition but still recommends it, 2 who were provided with this info but did not recommend the place, and 1 are not given information and did not recommended the restaurant.

Question 29.
Are there greater or fewer people that are informed of the nutrition of the food they eat?
Answer:
The goal is to determine if there are bigger or smaller population of people who are aware of the nutrition of the food they consume. Notice that the total number of people who answered no” when asked whether they have the nutritional information is greater than those who answered ‘yes’ Therefore, the are lesser number of people who are conscious of the nutrition of food they are taking.

Question 30.
Find the relative frequencies among the rows, and interpret their meanings. Round your answer to the nearest hundredth where necessary.
Answer:
The objective is to solve for the relative frequencies from the survey about preferred sports among the girls and boys.

To determine the relative frequency, divide each of the frequency of those who are aware of the nutritional information to the total number of those who recommend and do not recommend the restaurant Also, divide the frequency of those who are not provided with the nutritional information to the total number of those who recommend and do not recommend the restaurant.

From the table, observe that the highest reLative frequency obtained is the one where nutritional facts are not provided so the customers did not recommend the shop.
Therefore, when the owners refuse to give the nutritional information, the number of people who do not recommend the restaurant increases.

Question 31.
Find the relative frequencies among the columns, and interpret their meanings. Round your answer to the nearest hundredth where necessary.
Answer:
The objective is to solve for the relative frequencies from the survey about preferred sports among the girLs and boys.

To determine the relative frequency, divide each of the frequency of those who recommend the restaurant to the total number of those who are provided by nutritional information and to the total number of those who are not given the nutritional information. Also, divide the frequency of those who does not recommend the restaurant to the total number of those who are aware of nutritionaL information and to the total number of those who are not aware of the nutritional information.

Observe that the highest relative frequency is obtained is the one where nutritional facts are provided so the customers recommend the shop. Therefore, as the nutritional information is lay out by the restaurant owner, the number people who recommend the restaurant increases.

Question 32.
Math Journal Would you recommend that restaurant owners provide nutrition information for the menu items to their customers? Explain.
Answer:
The objective of this task is to make a recommendation on whether owners of restaurant lay out the nutritional facts of the food they serve. Since when the nutritional information is given by the restaurant owner, the number people who recommend the restaurant increases. On the other hand, when the owners refuse to give the nutritional information, the number of people who do not recommend the restaurant increases. Therefore, it is a good suggestion to the owners to give information about the food they serve for their customers to also recommend their shop to others.