### Guided Practice – Page No. 268

**Find the circumference of each circle.**

Question 1.

________ in

Answer: 56.57 in

Explanation:

Circumference of the circle = 2πr = 2 x 22/7 x 9 = 56.57 in

Question 2.

________ cm

Answer: 44 cm

Explanation:

Circumference of the circle = 2πr = 2 x 22/7 x 7 = 44 cm

**Find the circumference of each circle. Use 3.14 or \(\frac{22}{7}\) for π. Round to the nearest hundredth, if necessary.**

Question 3.

______ m

Question 4.

______ yd

Answer: 30.15 yd

Explanation:

Circumference of the circle = 2πr = 2 x 3.14 x 4.8 = 30.144 yd

Question 5.

______ in

Answer: 7.5 in

Explanation:

Circumference of the circle = 2πr = 2 x 3.14 x 7.5 = 47.1 in

Question 6.

A round swimming pool has a circumference of 66 feet. Carlos wants to buy a rope to put across the diameter of the pool. The rope costs $0.45 per foot, and Carlos needs 4 feet more than the diameter of the pool. How much will Carlos pay for the rope?

$ ______

Answer: $6.525

Explanation:

Circumference of the swimming pool = 66 feet

πd = 66

22/7 x d = 66

d = 66 x 7/ 22 = 10.5

The diameter of the pool = 10.5 feet

Carlos needs 4 feet more than the diameter of the pool.

Total rope needed = 10.5 + 4 = 14.5 feet

Cost of rope per foot = $0.45

Total cost of the rope = 14.5 x $0.45 = $6.525

Therefore the total cost of the rope = $6.525

**Find each missing measurement to the nearest hundredth. Use 3.14 for π.**

Question 7.

r =

d =

C = π yd

r = ________ yd

d = ________ yd

Answer:

r = 0.5 yd

d = 1 yd

Explanation:

Circumference = π yd

2πr = π yd

r = 1/2 yd = 0.5 yd

d = 2r = 2 [1/2] = 1 yd

Question 8.

r ≈

d ≈

C = 78.8 ft

r ≈ ________ ft

d ≈ ________ ft

Answer:

r = 495.31 ft

d = 990.62 ft

Explanation:

Circumference = 78.8 ft

2πr = 78.8 ft

r = 2 x 22/7 x 78.8 = 495.31 ft

d = 2 x 495.31 = 990.62 ft

Question 9.

r ≈

d ≈ 3.4 in

C =

r ≈ ________ in

C = ________ in

Answer:

r = 1.7 in

c = 10.68 in

Explanation:

Diameter = 3.4 in

Circumference = πd = 22/7 x 3.4 in = 10.68 in

r = d/2 = 1.7 in

**Essential Question Check-In**

Question 10.

Norah knows that the diameter of a circle is 13 meters. How would you tell her to find the circumference?

Type below:

____________

Answer: Circumference = 16.82 meters

Explanation:

Given,

Diameter = 13 meters

Circumference = πd = 22/7 x 13 = 16.82 meters

### Independent Practice – Page No. 269

**For 11–13, find the circumference of each circle. Use 3.14 or \(\frac{22}{7}\) for π. Round to the nearest hundredth, if necessary.**

Question 11.

_______ ft

Answer:

Cicumference = 18.526 ft = 19 ft (approx)

Explanation:

Given:

Diameter = 5.9 ft

Cicumference = πd = 3.14 x 5.9 = 18.526 ft = 19 ft (approx)

Question 12.

_______ cm

Answer:

Cicumference =176 cm

Explanation:

Given:

Radius = 56 cm

Cicumference = πd = 22/7 x 56 = 176 cm

Question 13.

_______ in

Answer:

Cicumference = 110 in

Explanation:

Given:

Diameter = 35 in

Cicumference = πd = 22/7 x 35 = 110 in

Question 14.

In Exercises 11–13, for which problems did you use \(\frac{22}{7}\) for π? Explain your choice.

Type below:

_____________

Answer:

11th question as 3.14 and the 12 and 13 questions as π

Explanation:

We can take 3.14 as π for 11 th question because the diameter is given in decimal points.

And in questions 12 and 13 we need to take π because the radius and diameter are given in whole number form.

Question 15.

A circular fountain has a radius of 9.4 feet. Find its diameter and circumference to the nearest tenth.

d = _________ ft

C = _________ ft

Answer:

d = 19 ft

C = 59 ft

Explanation:

Given:

Radius = 9.4 ft

Diameter = 2r = 2 x 9.4 = 18.8 ft = 19 ft (approx)

Circumference = πd = 22/7 x 18.8 = 59.08 = 59 ft (approx)

Question 16.

Find the radius and circumference of a CD with a diameter of 4.75 inches.

r = _________ in

C = _________ in

Answer:

r = 2.4 in

C = 15 in

Explanation:

Given:

Diameter = 4.75 in

Radius = r/2 = 4.75/2 = 2.37 in = 2.4 in (approx)

Circumference = πd = 22/7 x 4.75 = 14.92 in =15 in (approx)

Question 17.

A dartboard has a diameter of 18 inches. What are its radius and circumference?

r = _________ in

C = _________ in

Answer:

r = 9 in

C = 56.6 in

Explanation:

Given:

Diameter = 18 in

Radius = r/2 = 18/2 = 9 in

Circumference = πd = 22/7 x 18 = 56.57 in = 56.6 in (approx)

Question 18.

Multistep

Randy’s circular garden has a radius of 1.5 feet. He wants to enclose the garden with edging that costs $0.75 per foot. About how much will the edging cost? Explain.

$ _______

Answer:

Explanation:

Given:

The radius of the garden= 1.5 ft

Circumference of the garden = 2πr = 2 x 22/7 x 1.5 = 9.42 ft

Cost of enclosing the garden per foot = $0.75

Total cost of edging = 9.42 x $0.75 = $7.06 = $7 (approx)

Question 19.

Represent Real-World Problems

The Ferris wheel shown makes 12 revolutions per ride. How far would someone travel during one ride?

_______ ft

Answer: Total distance travelled in one ride is 4,752 ft

Explanation:

Given:

The diameter of the Ferris wheel= 63 ft

Circumference of the Ferris wheel = 2πr = 2 x 22/7 x 63 = 396 ft

Total number of revolutions = 12

Total distance travelled = 12 x 396 = 4,752 ft

Question 20.

The diameter of a bicycle wheel is 2 feet. About how many revolutions does the wheel make to travel 2 kilometres? Explain. Hint: 1 km ≈ 3,280 ft

_______ revolutions

Answer:

1044 revolutions

Explanation:

Given:

Diametre of the bicycle wheel = 2 feet

Total distance travelled = 2 kilometres

We know that,

1 km ≈ 3,280 ft

2 km = 2 x 3,280 = 6,560 ft

Circumference of the bicycle = Distance travelled in one revolution = πd = 22/7 x 2 = 6.28 ft = 6.3 ft

Total number of revolutions = Total distance travelled / distance travelled in one revolution

= 6560 / 6.28 = 1044 revolutions

Question 21.

Multistep

A map of a public park shows a circular pond. There is a bridge along a diameter of the pond that is 0.25 mi long. You walk across the bridge, while your friend walks halfway around the pond to meet you at the other side of the bridge. How much farther does your friend walk?

_______ mi

Answer:

Explanation:

Given,

The diameter of the pond = 0.25 mi

The length of the bridge = The diameter of the pond = 0.25 mi

Then the distance walked by the man = 0.25 mi

Distance travelled by the friend = Halfway around the pond to meet you at the other side of the bridge = πd/2

= 22/7 x 0.25/2 = 0.39 = 0.4 mi

The friend travelled more distance compared to the man

The more distance travelled by the friend = 0.39 – 0.25 = 0.14 mi

### Page No. 270

Question 22.

Architecture

The Capitol Rotunda connects the House and the Senate sides of the U.S. Capitol. Complete the table. Round your answers to the nearest foot.

Type below:

_____________

Answer:

Radius = 48 ft

Diameter = 96 ft

Explanation:

Given

Height = 180 ft

Circumference = 301.5 ft

πd = 301.5

22/7 x d = 301.5

d = 95.93 = 96 ft

r = d/2 = 96/2 = 48 ft

**H.O.T.**

**Focus on Higher Order Thinking**

Question 23.

Multistep

A museum groundskeeper is creating a semicircular statuary garden with a diameter of 30 feet. There will be a fence around the garden. The fencing costs $9.25 per linear foot. About how much will the fencing cost altogether?

$ _______

Answer:

The total cost of fencing = $712

Explanation:

Given,

The diameter = 30 ft

Circumference of the garden in the shape of circle = 2πr

Circumference of the semicircle = πr = πd/2 = 22/7 x 30/2 = 47.14ft

Cost of fencing for each foot = $9.25

The total cost of fencing the semicircular garden = 47.14 x $9.25 + 30 x $9.25 = $712 (approx)

Question 24.

Critical Thinking

Sam is placing rope lights around the edge of a circular patio with a diameter of 18 feet. The lights come in lengths of 54 inches. How many strands of lights does he need to surround the patio edge?

_______ strands

Answer: 12 and a half strands of light = 13 strands (approx)

Explanation:

Given,

The diameter of the circular patio = 18 ft = 216 inch

Circumference of the circular patio = πd = 22/7 x 216 = 678.85 inch

The lights will come in a length (in one strand)= 54 inches

Total number of strands of light required for the circular patio

= Circumference of the circular patio/ The lights will come in a length (in one strand) = 678.85/54 = 12.57 = 12 and a half strands of light

Question 25.

Represent Real-World Problems

A circular path 2 feet wide has an inner diameter of 150 feet. How much farther is it around the outer edge of the path than around the inner edge?

_______ feet

Answer: about 12.6 ft

Explanation:

Given,

Width of the circular path = 2 ft

The inner diameter of the circular path = 150 ft

The outer diameter of the circular path = 150 + 2(2) = 154 ft

Inner circumference = πd = 150 π

Outer circumference = πd = 154π

Distance between the outer and inner edge = 154 π – 150 π = 4 π = 12.6 ft

Question 26.

Critique Reasoning

Gear on a bicycle has the shape of a circle. One gear has a diameter of 4 inches, and a smaller one has a diameter of 2 inches. Justin says that the circumference of the larger gear is 2 inches more than the circumference of the smaller gear. Do you agree? Explain your answer.

_______

Answer:

Justin statement is incorrect.

Explanation:

The circumference of the larger gear = πd = 4π

The circumference of the smaller gear = πd = 2π

Since, 2 x 2π = 4π, the circumference of the larger gear is two times the circumference of the smaller gear.

Since = 4π – 2π = 2π = 6.28

Therefore, The larger circumference is not 2 inches more than the smaller circumference

Question 27.

Persevere in Problem Solving

Consider two circular swimming pools. Pool A has a radius of 12 feet, and Pool B has a diameter of 7.5 meters. Which pool has a greater circumference? How much greater? Justify your answers.

_______

Answer:

Pool B about 0.9 meters

Explanation:

Given,

Pool A has a diameter = 24 ft

Pool B has a diameter = 7.5 m

We know that,

1 ft = 0.3 metres

24 ft = 7.2 metres

The pool B has a greater diameter so it has a greater circumference.

Circumference of the pool A = 7.2π

Circumference of the pool B = 7.5π

Difference between the circumferences = 7.5π – 7.2π = 0.9 meters.

### Guided Practice – Page No. 274

**Find the area of each circle. Round to the nearest tenth if necessary. Use 3.14 for π.**

Question 1.

_______ m^{2}

Answer: 153.9 m^{2}

Explanation:

Given:

Diameter = 14 m

Radius = 14/2 = 7 m

Area of the circle = πr^{2}

= 3.14 x 7 x 7 = 153.86 = 153.9 m^{2}

Question 2.

_______ mm^{2}

Answer: 452.2 mm^{2}

Explanation:

Given:

Radius =12mm

Area of the circle = πr^{2}

= 3.14 x 12 x 12 = 3.14(144) = 452.2mm^{2}

Question 3.

_______ yd^{2}

Answer: 314 yd^{2}

Explanation:

Given:

Diameter = 20yd

Radius = 20/2 = 10yd

Area of the circle = πr^{2}

= 3.14 x 10 x 10 = 3.14(100) = 314yd^{2}

**Solve. Use 3.14 for π.**

Question 4.

A clock face has a radius of 8 inches. What is the area of the clock face? Round your answer to the nearest hundredth.

_______ in^{2}

Answer: 200.96 in^{2}

Explanation:

Given:

Radius = 8inches

Area of the clock face = πr^{2}

= 3.14 x 8 x 8= 3.14(64) = 200.96 in^{2}

Question 5.

A DVD has a diameter of 12 centimeters. What is the area of the DVD? Round your answer to the nearest hundredth.

_______ cm^{2}

Answer: 113.04 cm^{2}

Explanation:

Given:

Diameter = 12 centimeters

Radius = 12/2 = 6 centimeters

Area of the DVD= πr^{2}

= 3.14 x 6 x 6 = 3.14(36) = 113.04 cm^{2}

Question 6.

A company makes steel lids that have a diameter of 13 inches. What is the area of each lid? Round your answer to the nearest hundredth.

_______ in^{2}

Answer: 132.67 in^{2}

Explanation:

Given:

Diameter = 13 inches

Radius = 13/2 = 6.5 inches

Area of each lid= πr^{2}

= 3.14 x 6.5 x 6.5 = 3.14(42.25) = 132.67 in^{2}

**Find the area of each circle. Give your answers in terms of π.**

Question 7.

C = 4π

A =

Type below:

______________

Answer: 4π

Explanation:

Given:

Circumcenter = 4π

2πr = 4π

Radius = 4/2 = 2 units

Area of the circle = πr^{2}

= π x 2 x 2 = π(4) = 4π square units

Question 8.

C = 12π

A =

Type below:

______________

Answer: 36π

Explanation:

Given:

Circumcenter = 12π

2πr = 12π

Radius =6 units

Area of the circle = πr^{2}

= π x 6 x 6 = π(36) = 36π square units

Question 9.

C = \(\frac{π}{2}\)

A =

Type below:

______________

Answer: π/16

Explanation:

Given:

Circumcenter = \(\frac{π}{2}\)

2πr = \(\frac{π}{2}\)

Radius = 1/4 units

Area of the circle = πr^{2}

= π x 1/4 x 1/4 = π(1/16) = π/16 square units

Question 10.

A circular pen has an area of 64π square yards. What is the circumference of the pen? Give your answer in terms of π

Type below:

______________

Answer: 16π

Explanation:

Given:

Area of the circular pen = 64π square yards

πr^{2} = 64π

r = 8 yards

Circumference of the circle = 2πr = 2 x 8 x π = 16π yards

**Essential Question Check-In**

Question 11.

What is the formula for the area A of a circle in terms of the radius r?

Type below:

______________

Answer: πr^{2}

Explanation:

Area of a circle = πr^{2}

### Independent Practice – Page No. 275

Question 12.

The most popular pizza at Pavone’s Pizza is the 10-inch personal pizza with one topping. What is the area of a pizza with a diameter of 10 inches? Round your answer to the nearest hundredth.

_______ in^{2}

Answer: 78.5 in^{2}

Explanation:

Given:

Diameter = 10 inches

Radius = 10/2 = 5 inches

Area of a pizza = πr^{2}

= 3.14 x 5 x 5 = 3.14(25) = 78.5 in^{2}

Question 13.

A hubcap has a radius of 16 centimeters. What is the area of the hubcap? Round your answer to the nearest hundredth.

_______ cm^{2}

Answer: 803.84 cm^{2}

Explanation:

Given:

Radius = 16 cm

Area of the circle = πr^{2}

= 3.14 x 16 x 16 = 3.14(256) = 803.84 cm^{2}

Question 14.

A stained glass window is shaped like a semicircle. The bottom edge of the window is 36 inches long. What is the area of the stained glass window? Round your answer to the nearest hundredth.

_______ in^{2}

Answer: 508.68 in^{2}

Explanation:

Area of the semicircle = 1/2 πr^{2} = 1/2(3.14)(18)(18) = 1/2 (3.14)(324) = 1.57(324) = 508.68 in^{ 2 }

Question 15.

Analyze Relationships

The point (3,0) lies on a circle with the centre at the origin. What is the area of the circle to the nearest hundredth?

_______ units^{2}

Answer: 28.26 units^{2}

Explanation:

Radius = 3

Area of the circle = πr^{2} = π(3)^{2} = 3.14(9) = 28.26 units^{2}

Question 16.

Multistep

A radio station broadcasts a signal over an area with a radius of 50 miles. The station can relay the signal and broadcast over an area with a radius of 75 miles. How much greater is the area of the broadcast region when the signal is relayed? Round your answer to the nearest square mile.

_______ mi^{2}

Answer: 9813 mi^{2}

Explanation:

Given:

The radius of a radio station broadcasted the signal (r) = 50 miles

The greatest radius to which the broadcast can be relayed (R) = 75 miles

The greatest area of the broadcast region when the signal is relayed = πR^{2}-πr^{2} = π(75) (75) – π (50) (50)

= 5625π – 2500π

= 3125π

= 3125(3.14) = 9813 mi^{2}(approx)

Question 17.

Multistep

The sides of a square field are 12 meters. A sprinkler in the center of the field sprays a circular area with a diameter that corresponds to a side of the field. How much of the field is not reached by the sprinkler? Round your answer to the nearest hundredth.

_______ m^{2}

Answer:30.96 m^{2}

Explanation:

Given:

The side of the square = 12 meters

The diameter circular area of the field in the centre = The side of the square = 12 meters

The radius of the field = 12/2 = 6 meters

Area of the field which is not reached by the sprinkler = Area of the square – Area of the circular area

= (side)^{2}-πr^{2} = (12)(12) – π (6) (6)

= 144 – 36 (3.14)

= 144 – 113.04

= 30.96 m^{2}

Question 18.

Justify Reasoning

A small silver dollar pancake served at a restaurant has a circumference of 2π inches. A regular pancake has a circumference of 4π inches. Is the area of the regular pancake twice the area of the silver dollar pancake? Explain.

_______

Answer: No, the area of the regular pancake is 4 times the area of the silver dollar pancake

Explanation:

Silver Dollar pancake:

Circumference of the silver Dollar pancake = 2π inches

2πr = 2π

r = 1 inch

Area of the silver dollar pancake = πr^{2} = π (1) (1) = π in^{2}

Regular pancake:

Circumference of the regular pancake = 4π inches

2πr = 4π

r = 2 inch

Area of the silver dollar pancake = πr^{2} = π (2) (2) = 4π in^{2}

Therefore, the area of the regular pancake is 4 times the area of the silver dollar pancake

Question 19.

Analyze Relationships

A bakery offers a small circular cake with a diameter of 8 inches. It also offers a large circular cake with a diameter of 24 inches. Does the top of the large cake have three times the area of that of the small cake? If not, how much greater is its area? Explain.

_______

Answer: No, the area of the large cake is 9 times the area of the small cake

Explanation:

Small Cake:

The diameter of the small cake= 8 inches

The radius of the small cake = 8/2 = 4 inches

Area of the small cake = πr^{2} = π (4) (4) = 16 π in^{2}

Large Cake:

The diameter of the large cake= 24 inches

The radius of the large cake = 24/2 = 12 inches

Area of the large cake = πr^{2} = π (12) (12) = 144 π in^{2
}

Since 144 π/ 16 π = 9

Therefore the

area of the large cake is 9 times the area of the small cake.

### Page No. 276

Question 20.

Communicate Mathematical Ideas

You can use the formula A = \(\frac{C^{2}}{4π}\) to find the area of a circle given the circumference. Describe another way to find the area of a circle when given the circumference.

Type below:

____________

Answer: Area = C^{2}/4π

Explanation:

Circumference of the circle = 2πr

C = 2πr

Divide both sides by 2π

then, r = C/2π

Area of the circle = πr^{2}

Substitute C/2π for r:

Area = π(c/2π)^{2} = C^{2}/4π

Question 21.

Draw Conclusions

Mark wants to order a pizza. Which is the better deal? Explain.

_____

Answer: The pizza of 18 inches is a better deal

Explanation:

Given:

The diameter of the pizza = 12 inches

The radius of the pizza = 12/2= 6 inches

Area of the circle = πr^{2}

= (3.14)(6)(6) = 113 (approx) in^{2}

The total cost of the pizza = $10

Cost of the pizza per inch = $10/113 = $0.09 per square inch

The diameter of the pizza = 18 inches

The radius of the pizza = 18/2= 9 inches

Area of the circle = πr^{2}

= (3.14)(9)(9) = 254 (approx) in^{2}

The total cost of the pizza = $20

Cost of the pizza per inch = $20/254 = $0.08 per inch

Question 22.

Multistep

A bear was seen near a campground. Searchers were dispatched to the region to find the bear.

a. Assume the bear can walk in any direction at a rate of 2 miles per hour. Suppose the bear was last seen 4 hours ago. How large an area must the searchers cover? Use 3.14 for π. Round your answer to the nearest square mile.

_____ mi^{2}

Answer: 201mi^{2}

Explanation:

The bear can walk a distance = 2 x 4 = 8 miles

Since it is walking 2 miles per hour for 4 hours

The radius of the bear = 8 miles

Area of the circle = πr^{2}

= (3.14)(8)(8) = 201 (approx) mi^{2}

Question 22.

b. What If? How much additional area would the searchers have to cover if the bear were last seen 5 hours ago?

_____ mi^{2}

Answer: 113mi^{2}

Explanation:

If the bear for 5 hours then,

The bear can walk a distance = 2 x 5 = 10 miles

Since it is walking 2 miles per hour for 5 hours

The radius of the bear = 10 miles

Area of the circle = πr^{2}

= (3.14)(10)(10) = 314 (approx) mi^{2}

The additional area covered by the searches = 314 – 201 = 113 mi^{2}

**H.O.T.**

**Focus on Higher Order Thinking**

Question 23.

Analyze Relationships

Two circles have the same radius. Is the combined area of the two circles the same as the area of a circle with twice the radius? Explain.

_____

Answer: No

Explanation:

If the radius of two circles is the same.

then, Let the radii of the circles be 1.

The area of each circle = π square units

The combined area of 2 circles =π+π = 2π square units

If the radius is doubled.

then, Let the radii of the circles be 2

The area of each circle = 4π square units

The combined area of 2 circles = 4π+4π = 8π square units

Therefore the areas of both cases are not the same.

Question 24.

Look for a Pattern

How does the area of a circle change if the radius is multiplied by a factor of n, where n is a whole number?

Type below:

____________

Answer: The new area is then n^{2} times the area of the original circle.

Explanation:

If the radius is multiplied by a factor “n”

then, the new radius = rn

The area of the circle (with radius rn) = π(rn)^{2 }= n^{2} (πr^{2}).

Therefore the new area is n^{2} times the area of the original circle.

Question 25.

Represent Real World Problems

The bull’s-eye on a target has a diameter of 3 inches. The whole target has a diameter of 15 inches. What part of the whole target is the bull’s-eye? Explain.

Type below:

____________

Answer: 1/25 of the target

Explanation:

Bull’s eye:

Diameter of Bull’s eye = 3 inches

Radius of Bull’s eye = 3/2 = 1.5 inches

Area of the Bull’s eye = π(r)^{2 }= π(1.5)^{2} = 2.25π

Target:

Diameter of the target = 15 inches

Radius of the target = 15/2 = 7.5 inches

Area of the target = π(r)^{2 }= π(7.5)^{2} = 56.25π

The part of Bull’s eye in the whole target = 2.25π/ 56.25π = 1/25

Therefore the 1/25th part of the whole target is the Bull’s eye.

### Guided Practice – Page No. 280

Question 1.

A tile installer plots an irregular shape on grid paper. Each square on the grid represents 1 square centimeter. What is the area of the irregular shape?

_____ cm^{2}

Answer: Area of the irregular shape = 34 cm^{2}

Explanation:

STEP1 First divide the irregular shapes into polygons.

STEP2 The irregular shape can be divided into a triangle, rectangle, parallelogram

STEP3 Areas of the polygons

Area of triangle = 1/2 (base x height) = 1/2 (4 x 2) = 4 cm^{2}

Area of the rectangle = length x breadth = 5 x 3 = 15 cm^{2}

Area of the parallelogram = base x height = 5 x 3 = 15 cm^{2}

Area of the irregular shape = (15+15+5) cm^{2}= 34cm^{2}

Question 2.

Show two different ways to divide the composite figure. Find the area both ways. Show your work below.

_____ cm^{2}

Answer: Area of the figure in both ways = 288 cm^{2}

Explanation:

The first way to divide up the composite shape is to divide it into an 8 by 9 rectangle and a 12 by 18 rectangle.

The area of the first rectangle = Length x breadth = 9 x 8 = 72 cm^{2}

The area of the second rectangle = Length x breadth = 18 x 12 = 216 cm^{2}

The total area of the figure = 72 + 216 = 288 cm^{2}

Question 3.

Sal is tiling his entryway. The floor plan is drawn on a unit grid. Each unit length represents 1 foot. Tile costs $2.25 per square foot. How much will Sal pay to tile his entryway?

$ _____

Answer: Sal will pay $97.875

Explanation:

Separate this figure into trapezium and parallelogram.

Area of the trapezium = 1/2 (a+b)h = 1/2 (7+4) 5 = 1/2 (11) 5 = 27.5 ft^{2}

Area of the parallelogram = base x height = 4 x 4 = 16 ft^{2}

The total area of the figure = 27.5 + 16 = 43.5ft^{2}

Cost of each square foot = $2.25

Amount paid by Sal = 43.5 x 2.25 = $97.875

**Essential Question Check-In**

Question 4.

What is the first step in finding the area of a composite figure?

Type below:

______________

Answer:

The first step in finding the area of a composite figure is to divide it up into smaller basic shapes.

Explanation:

The first step in finding the area of a composite figure is to divide it up into smaller basic shapes such as triangles, squares, rectangles, parallelograms, circles and trapezium.

Then calculate the area of each figure and add them to find the area of the figure.

### Independent Practice – Page No. 281

Question 5.

A banner is made of a square and a semicircle. The square has side lengths of 26 inches. One side of the square is also the diameter of the semicircle. What is the total area of the banner? Use 3.14 for π.

_____ in^{2}

Answer: 941.33 in^{2}

Explanation:

Area of the square = side x side = 26 x 26 = 676 in^{2}

Area of the semicircle =1/2 πr^{2}= 1/2 (3.14) (13) (13) = 1/2 (3.14) (169) = 265.33 in^{2}

Area of the figure = 676 + 265.33 = 941.33 in^{2}

Question 6.

Multistep

Erin wants to carpet the floor of her closet. A floor plan of the closet is shown.

a. How much carpet does Erin need?

_____ ft^{2}

Answer: 61 ft^{2}

Explanation:

Area of the rectangle = length x breadth = 4 x 10 = 40 ft

Area of the triangle = 1/2 x base x height = 1/2 x 6 x 7 = 21 ft

The total area of the figure = 40+21 = 61 ft^{2}

Question 6.

b. The carpet Erin has chosen costs $2.50 per square foot. How much will it cost her to carpet the floor?

$ _____

Answer: $152.50

Explanation:

Cost per square foot of the carpet = $2.50

The total cost of the carpet on the floor = 61 x $2.50 =$152.50

Question 7.

Multiple Representations

Hexagon ABCDEF has vertices A(-2, 4), B(0, 4), C(2, 1), D(5, 1), E(5, -2), and F(-2, -2). Sketch the figure on a coordinate plane. What is the area of the hexagon?

_____ units^{2}

Answer: The area of the figure is 30 square units

Explanation:

Separate the figure into a trapezium and a rectangle.

Area of a trapezium = 1/2 (a+b) h= 1/2 (2+4) x 3 = 1/2 (6) 3 = 9 square units

Area of a rectangle = length x breadth = 7 x 3 = 21 square units

The total area of the figure = 9+21 = 30 square units

Question 8.

A field is shaped like the figure shown. What is the area of the field? Use 3.14 for π.

_____ m^{2}

Answer: 146.24 m^{2}

Explanation:

Divide the figure into a square, triangle and a quarter of a circle.

Area of a square = side x side = 8 x 8 = 64 m^{2}

Area of a quarter of a circle = 1/4 (πr^{2}) = 1/4 (3.14 x 8^{2})

= 1/4 (200.96) = 50.24 m^{2}

Area of the triangle = 1/2 x base x height = 1/2 x 8 x 8 = 32 m^{2}

Total area of the figure = 64+32+50.24 = 146.24 m^{2}

Question 9.

A bookmark is shaped like a rectangle with a semicircle attached at both ends. The rectangle is 12 cm long and 4 cm wide. The diameter of each semicircle is the width of the rectangle. What is the area of the bookmark? Use 3.14 for π.

_____ cm^{2}

Answer: 60.56 cm^{2}

Explanation:

The bookmark is divided into a rectangle, semicircle.

Area of the rectangle = length x breadth = 12 x 4 = 48 cm^{2}

The diameter of the semicircle = The width of the rectangle = 4 cm

The radius of the semicircle = 4/2 = 2 cm

The area of the semicircle = πr^{2} = 3.14 x 2 x 2 = 12.56 cm^{2}

The total area of the bookmark = 12.56 + 48 = 60.56 cm^{2}

Question 10.

Multistep

Alex is making 12 pendants for the school fair. The pattern he is using to make the pennants is shown in the figure. The fabric for the pennants costs $1.25 per square foot. How much will it cost Alex to make 12 pennants?

$ _____

Answer: $52.50

Explanation:

Each pendant is made up of a rectangle and a triangle.

Area of the rectangle = length x breadth = 3 x 1 = 3 ft^{2}

Area of the triangle = 1/2 x base x height = 1/2 x 1 x 1 = 0.5 ft^{2}

The total area of the pendant = 3+0.5 = 3.5 ft^{2}

Number of pendants = 12

Area of the pendants = 12 x 3.5 = 42 ft^{2}

Cost of each square feet of the pendant = $1.25

Total cost for all the 12 pendants = 12 x $1.25 = $52.50

Question 11.

Reasoning

A composite figure is formed by combining a square and a triangle. Its total area is 32.5 ft^{2}. The area of the triangle is 7.5 ft^{2}. What is the length of each side of the square? Explain.

_____ ft

Answer: 5 ft

Explanation:

Given:

The area of the composite figure = 32.5 ft^{2}

The area of the triangle = 7.5 ft^{2}

The area of the square = 32.5 – 7.5 = 25

side x side = 25

side^{2} = 25

side = root 25 = 5 ft

### H.O.T. – Page No. 282

**Focus on Higher Order Thinking**

Question 12.

Represent Real-World Problems

Christina plotted the shape of her garden on graph paper. She estimates that she will get about 15 carrots from each square unit. She plans to use the entire garden for carrots. About how many carrots can she expect to grow? Explain.

______ carrots

Answer: 300 carrots

Explanation:

This shape is divided into two triangles and a square.

Area of figure = 2(1/2 x 2 x 2) + 4(4) = 4 + 16 = 20 square units

Number of carrots per square unit = 300

Total number of carrots = 20 x 15 = 300

Question 13.

Analyze Relationships

The figure shown is made up of a triangle and a square. The perimeter of the figure is 56 inches. What is the area of the figure? Explain.

_____ in^{2}

Answer: 192 in^{2}

Explanation:

Given:

The perimeter of the figure = 56 inches

The figure is divided into a square and a triangle.

10 + 10 + 3s = 56

3s = 36

s = 12

The area of a triangle = 1/2 x 12 x 8 = 48 in^{2}

The area of a square = 12 x 12 = 144 in^{2}

Total area of the figure = 144 + 48 = 192 in^{2}

Question 14.

Critical Thinking

The pattern for a scarf is shown at right. What is the area of the scarf? Use 3.14 for π.

_____ in^{2}

Answer: 243 in^{2}

Explanation:

Area of the rectangle in the given figure = 28 x 15 = 420 in^{2}

Area of two semicircles = 2 (1/2 πr^{2} ) = 3.14 x 7.5 x 7.5 = 176.625 in^{2}

Area of the shaded region = 420 – 176.625 = 243 in^{2}(approx)

Question 15.

Persevere in Problem Solving

The design for the palladium window shown includes a semicircular shape at the top. The bottom is formed by squares of equal size. A shade for the window will extend 4 inches beyond the perimeter of the window, shown by the dashed line around the window. Each square in the window has an area of 100 in^{2}.

a. What is the area of the window? Use 3.14 for π.

_____ in^{2}

Answer: a) 2228 in^{2}

Explanation:

Area of the square = 100 in^{2}

side x side = 100

Side = 10 in

Since the side of each square is 10 in and there are 4 squares.

The side length of the larger square (s) = 40 in

Area of the larger square = side x side = 40 x 40 = 1600 in^{2}

Since the side of each square is 10 in and there are 2 squares.

The radius of the semi-circle = 20 in

Area of the semi-circle = 1/2(πr^{2}) = 1/2(3.14 x 20^{2}) = 628 in^{2}

The area of the window = 1600 + 628 = 2228 in^{2}

Question 15.

b. What is the area of the shade? Round your answer to the nearest whole number.

_____ in^{2}

Answer: b) 3016 in^{2}

Explanation:

The shade extends 4 inches beyond the shapes so the length of the bottom rectangle is 40+4+4 = 48 in

The length extends below the original square.

The height is now = 40+4 = 44 in

The radius of the semi-circle = 20+4 = 24 in

The new area of the figure = 48(44) + 1/2(3.14 x 24^{2}) = 2112 + 904.32 = 3016.32 = 3016 in^{2}

### Guided Practice – Page No. 286

**Find the surface area of each solid figure.**

Question 1.

Total surface area: _____ ft^{2}

Answer: 150 ft^{2}

Explanation:

The base is a triangle with side lengths of 8 ft, 5 ft, 5 ft so the perimeter of the base = P = 8+5+5 = 18 ft

The height of the prism = 7 ft

The base is a triangle.

Area of the triangle = 1/2 (8) (3) = 12 ft^{2}

The surface area formula for a prism is S = Ph + 2b

P = Perimeter = 18 h = height = 7 b = base = area of the triangle = 12

The surface area of the prism = 18(7) + 2(12) = 126 + 24 = 150 ft^{2}

Question 2.

Total surface area: _____ m^{2}

Answer: 503 m^{2}

Explanation:

Given:

Dimensions of the cuboid:

Length = 11 m

Breadth = 9 m

Height = 7 m

The surface area of the cuboid = 2(lb+bh+hl) = 2(11 x 9 + 9 x 7 + 7 x 11) = 478m^{2
}

The dimensions of the cube:

Length of the side = 2.5 m

The surface area of the cube = 6a^{2} = 6 x 2.5 x 2.5 = 37.5 m^{2}

The surface area of the rectangular prism = 2.5 x 2.5 = 6.25

The surface area of the figure = The overlapping area is the area of the base of the cube

= 37.5 + 478 – 2(6.25) = 503 m^{2}

**Essential Question Check-In**

Question 3.

How can you find the surface area of a composite solid made up of prisms?

Type below:

_____________

Answer: The surface area of the prisms, add them up, and then subtract the overlapping areas twice.

Explanation:

The surface area of a composite solid is made up of prisms by finding the surface areas of the prisms, adding them up, and then up, and then subtracting the overlapping areas.

### Independent Practice – Page No. 287

Question 4.

Carla is wrapping a present in the box shown. How much wrapping paper does she need, not including overlap?

_____ in^{2}

Answer: 164 in^{2}

Explanation:

The surface area of the cuboid excluding the top = 2h(l+b) + lb = 2 x 4 ( 13 ) + 10 x 3 = 164 in^{2}

The length of the wrapping paper = The surface area of the cuboid excluding the top = 164 in^{2}

Question 5.

Dmitri wants to cover the top and sides of the box shown with glass tiles that are 5 mm square. How many tiles does he need?

_____ tiles

Answer: 3720 tiles

Explanation:

The surface area of the cuboid excluding the bottom = 2h(l+b) + lb = 2 x 9 (35) + 20 x 15 = 930 cm^{2}

5mm = 0.5 cm

Area of a tile = Area of the square = a^{2} = 0.5cm x 0.5cm = 0.25 cm^{2}

Total number of tiles = 930/0.25 = 3720 tiles

Question 6.

Shera is building a cabinet. She is making wooden braces for the corners of the cabinet. Find the surface area of each brace.

_____ in^{2}

Answer: 45 in^{2}

Explanation:

The perimeter of the figure = P = 3(3) + 2(1) = 11 in

Base = B = 3(2) = 6 in

Height = h = 3

The surface area of the figure = Ph + 2B = 11 x 3 +2(6) = 33 + 12 = 45 in^{2}

Question 7.

The doghouse shown has a floor, but no windows. Find the total surface area of the doghouse, including the door.

_____ ft^{2}

Answer: 66ft^{2}

Explanation:

Perimeter of the pentagon base (P) = 2(2.5) + 2(2) + 3 = 5 + 4 + 3 = 12

Area of the pentagon base by adding the area of the triangle and the area of the rectangle (B) = 1/2(3)(2) + 2(3) = 9

Height (h) = 2 + 2 = 4

The surface area of the figure = Ph + 2B = 12(4) + 2(9) = 48 + 18 = 66ft^{2}

**Eddie built the ramp shown to train his puppy to do tricks. Use the figure for 8–9.**

Question 8.

Analyze Relationships

Describe two ways to find the surface area of the ramp.

Type below:

____________

Answer: One way is to use the formula S = Ph + 2B. Another way is to find the area of each face of the prism and add them up to get the total surface area.

Explanation:

The very first way to use the formula S = Ph + 2B where the trapeziums are the base. The second way is to find the area of each face of the prism and then add them up to get the total surface area.

Question 9.

What is the surface area of the ramp?

_____ in^{2}

Answer: 3264 in^{2}

Explanation:

P = Perimeter of the figure = 16(3) + 2 (20) + 16 = 104

B = Base of the figure = 1/2 (12) (16 + 3(16)) = 6 (16 + 48) = 6 (64) = 384

h = Height of the figure = 2

Surface area of the figure = Ph + 2B = 104(2) + 2(384) = 2496 + 768 = 3264 in^{2}

**Marco and Elaine are building a stand like the one shown to display trophies. Use the figure for 10–11.**

Question 10.

What is the surface area of the stand?

_____ ft^{2}

Answer: 58 ft^{2}

Explanation:

Top:

Perimeter = P = 4(1) = 4

Base = B = 1(1) = 1

Height = h = 3

Top surface area = Ph + 2B = 4(3) + 2(1) = 14 ft^{2}

Bottom :

Perimeter = P = 2(7) + 2(1) = 14 + 2 = 16

Base = B = 7(1) = 7

Height = h = 2

Top surface area = Ph + 2B = 16(2) + 2(7) = 46 ft^{2}

Overlapping area = 1(1) = 1

The surface area of the figure = The surface area of the top + The surface area of the bottom – the overlapping area = 14 + 46 – 2 = 60 – 2 = 58 ft^{2}

Question 11.

Critique Reasoning

Marco and Elaine want to paint the entire stand silver. A can of paint covers 25 square feet and costs $6.79. They set aside $15 for paint. Is that enough? Explain.

_____

Answer: No

Explanation:

Since the surface area is 58 ft^{2}, they will need 3 cans of paint. Since each can paints 25 ft^{2} and we cannot buy a fraction of cans.

3 cans would then cost 6.79 x 3 = 20.37 so this is not enough.

### Page No. 288

Question 12.

Henry wants to cover the box shown with paper without any overlap. How many square centimeters will be covered with paper?

_____ cm^{2}

Answer: 2316 cm^{2}

Explanation:

Given:

Length = 24cm Breadth = 27cm Height = 10cm

P = Perimeter = 2(24) + 2(27) = 48 + 54 = 102

B = Base = 24(27) = 648

h = Height = 10

Surface area of the figure = Ph + 2B = 102(10) + 2(648) = 1020 + 1296 = 2316 cm^{2}

Question 13.

What If?

Suppose the length and width of the box in Exercise 12 double. Does the surface area S double? Explain.

_____

Answer: No

Explanation:

Given :

Length = 24cm x 2 = 48 cm Breadth = 27cm x 2 = 54 cm Height = 10cm

P = 2(48) + 2(54) = 96 + 108 = 204

B = 48(54) = 2592

New Surface area = Ph + 2B = 204(10) + 2(2592) = 2040 + 5184 = 7224 cm^{2}

Double of surface area = 2 (2316) = 4632 cm^{2}

So the new surface area is not double of the initial area.

**H.O.T.**

**Focus on Higher Order Thinking**

Question 14.

Persevere in Problem Solving

Enya is building a storage cupboard in the shape of a rectangular prism. The rectangular prism has a square base with side lengths of 2.5 feet and a height of 3.5 feet. Compare the amount of paint she would use to paint all but the bottom surface of the prism to the amount she would use to paint the entire prism.

Type below:

______________

Answer: The difference would just be the area in the bottom surface. It would be 6.25 ft^{2} less.

Explanation:

The difference in the amount of paint would just be the area of the bottom surface. The area of the bottom surface is (2.5)^{2} = 6.25.

Therefore she would paint 6.25 ft^{2} less if she painted all but the bottom surface compared to painting the entire prism.

Question 15.

Interpret the Answer

The oatmeal box shown is shaped like a cylinder. Use a net to find the surface area S of the oatmeal box to the nearest tenth. Then find the number of square feet of cardboard needed for 1,500 oatmeal boxes. Round your answer to the nearest whole number

_____ ft^{2}

Answer: 138.28 in^{2} , 1440 ft^{2}

Explanation:

Given:

Dimensions of the cylinder:

Radius: 2 in

Height: 9 in

The total surface area of the cylinder = 2πr(r+h) = 2 x 22/7 x 2 (2 + 9) = 138.28 in^{2}

The total number of square inches needed for 1,500 oatmeal boxes = 1,500 x 138.28 = 207,300 in^{2}

1 ft = 12 in

(1 ft)^{2} = (12 in)^{2}

1 ft^{2} = 144 in^{2}

The total number of square feet needed for 1,500 oatmeal boxes (to the nearest whole number)

= 207,300/144 = 1440 ft^{2}

Question 16.

Analyze Relationships

A prism is made of centimeter cubes. How can you find the surface area of the prism in Figure 1 without using a net or a formula? How does the surface area change in Figures 2, 3, and 4? Explain.

Type below:

______________

Answer: The surface area for the first 3 figures are the same. The surface area for figure 4 is greater than the surface area of the figures 1 – 3.

Explanation:

The surface area of the first 3 figures is the same. The 3 new faces on figure 2 have the same areas as the 3 visible faces that were removed when the top corner cube was removed. The surface area is then the same as it is for figure 1. Similarly, the areas of the new visible faces in figure 3 are equal to the areas of the visible faces removed from removing the corner cubes so the surface areas are the same as in figure 1. The surface area for figure 4 is greater than the surface areas of the figures 1 – 3. Removing the cube removed 2 of the visible faces (one from the top and one from the front side) but added 4 visible faces so the surface area increases.

### Guided Practice – Solving Volume Problems – Page No. 292

Question 1.

Find the volume of the triangular prism.

_____ ft^{3}

Answer: 84 ft^{3}

Explanation:

Base area of the prism = 1/2 x 8 x 3 = 12 ft^{2}

Height of the prism = 7 ft

Volume of the prism = (12 x 7) ft^{3}

Question 2.

Find the volume of the trapezoidal prism.

_____ m^{3}

Answer: 330 m^{3}

Explanation:

Base area of the prism = 1/2 x (15 + 5) x 3 = 30 m^{2}

Height of the prism = 11 m

Volume of the prism = (30 x 11) m^{3} = 330 m^{3}

Question 3.

Find the volume of the composite figure.

_____ ft^{2}

Answer: Composite figure: 360 ft^{3}

Explanation:

The volume of the triangular prism:

The base area of the prism = 1/2 x 4 x 6 = 12 ft^{2}

Height = 6 ft

The volume of the triangular prism = 12 x 6 = 72 ft^{3}

The volume of the rectangular prism:

The base area of the prism = 4 x 6 = 24 ft^{2}

Height = 12 ft

The volume of the triangular prism = 12 x 24 = 288 ft^{3}

Volume of the composite figure = (288 + 72)ft^{3} = 360 ft^{3}

**Find the volume of each figure.**

Question 4.

The figure shows a barn that Mr. Fowler is building for his farm.

_____ ft^{3}

Answer: 40,000 ft^{3}

Explanation:

Triangular prism:

B = Base area = 1/2 x 10 (40) = 200 cm^{2}

Height = 50 cm

The volume of the triangular prism = Bh = 200 x 50 = 10,000 cm^{3}

Rectangular prism:

B = Base area =40 x 15 = 600 cm^{2}

Height = 50 cm

The volume of the triangular prism = Bh = 600 x 50 = 30,000 cm^{3}

Total volume of the prism = 10,000 + 30,000 = 40,000 cm^{3}

Question 5.

The figure shows a container, in the shape of a trapezoidal prism, that Pete filled with sand.

_____ cm^{3}

Answer: 385 cm^{3}

Explanation:

B = Base area = 1/2 x 5 (10 + 12) = 55 cm^{2}

Height = 7 cm

The volume of the container = Bh = 55 x 55 = 385 cm^{3}

**Essential Question Check-In**

Question 6.

How do you find the volume of a composite solid formed by two or more prisms?

Type below:

______________

Answer: Finding the volume of each figure adding them up to get the volume of the composite solid.

Explanation:

To find the volume of the composite figure that can be divided into 2 or more prisms, find the volume of each prism and add them up to get the volume of the composite solid.

### Independent Practice – Page No. 293

Question 7.

A trap for insects is in the shape of a triangular prism. The area of the base is 3.5 in^{2} and the height of the prism is 5 in. What is the volume of this trap?

_____ in^{3}

Answer: 17.5 in^{3}

Explanation:

The volume of the trap = Base area x height = 3.5 x 5 = 17.5 in^{3}

Question 8.

Arletta built a cardboard ramp for her little brothers’ toy cars. Identify the shape of the ramp. Then find its volume.

Shape: _________

Area: _________ in^{3}

Answer: 525 in^{3}

Explanation:

Base area = 1/2 x 6 x 25 = 75 in^{2}

Height = 7 in

Volume of the figure = 75 x 7 = 525 in^{3}

Question 9.

Alex made a sketch for a homemade soccer goal he plans to build. The goal will be in the shape of a triangular prism. The legs of the right triangles at the sides of his goal measure 4 ft and 8 ft, and the opening along the front is 24 ft. How much space is contained within this goal?

_____ ft^{3}

Answer: 384 ft^{3}

Explanation:

Base area = 1/2 x 4 x 8 = 16 ft^{2}

Height = 24 ft

Volume of the figure = 16 x 24 = 384 ft^{3}

Question 10.

A gift box is in the shape of a trapezoidal prism with base lengths of 7 inches and 5 inches and a height of 4 inches. The height of the gift box is 8 inches. What is the volume of the gift box?

_____ in^{3}

Answer: 192 in^{3}

Explanation:

Base area = 1/2 x 4 x (7+5) = 24 in^{2}

Height = 8 in

Volume of the figure = 24 x 8 = 192 Base area = 1/2 x 6 x 25 = 75 in^{2}

Height = 7 in

Volume of the figure = 75 x 7 = 525 in^{3}

Question 11.

Explain the Error

A student wrote this statement: “A triangular prism has a height of 15 inches and a base area of 20 square inches. The volume of the prism is 300 square inches.” Identify and correct the error.

Type below:

____________

Answer: The error is measurement unit.

Explanation:

The volume of the prism is:

base area x height = 20 x 15 = 300 in^{3}

**Find the volume of each figure. Round to the nearest hundredth if necessary.**

Question 12.

_____ in^{3}

Answer: 97.2 in^{3}

Explanation:

The volume of the hexagonal prism = 23.4 x 3 = 70.2 in^{3}

Base area of the rectangular prism = 3 x 3 = 9 in^{2}

The volume of the rectangular prism = Bh = 9 x 3 = 27 in^{3}

Total volume of the figure = 70.2 + 27 = 97.2 in^{3}

Question 13.

_____ m^{3}

Answer: 316.41 m^{3}

Explanation:

The volume of the rectangular prism on the left = Bh = [7.5 x 3.75] (3.75) = 105.47 m^{3}

The volume of the rectangular prism on the right = Bh = [7.5 x 3.75](7.5) = 210.94 m^{3}

Total volume of the composite figure = 105.47 + 210.94 = 316.41 m^{3}

Question 14.

Multi-Step

Josie has 260 cubic centimeters of candle wax. She wants to make a hexagonal prism candle with a base area of 21 square centimeters and a height of 8 centimeters. She also wants to make a triangular prism candle with a height of 14 centimeters. Can the base area of the triangular prism candle be 7 square centimeters? Explain.

_____

Answer: No

Explanation:

The volume of the hexagonal prism = 21 x 8 = 168

The total volume of wax, 260 is equal to the sum of the volumes of each prism.

B is the base area of the triangular prism.

168 + 14B = 260 cm^{3}

14B = 260 – 168

B = 6.6 cm^{3}

### Page No. 294

Question 15.

A movie theater offers popcorn in two different containers for the same price. One container is a trapezoidal prism with a base area of 36 square inches and a height of 5 inches. The other container is a triangular prism with a base area of 32 square inches and a height of 6 inches. Which container is the better deal? Explain.

Type below:

___________

Answer: The triangular prism is a better deal since it has a larger volume

Explanation:

The base area of the trapezoidal prism = 36 in^{2}

The volume of the trapezoidal prism = Bh = 36 x 5 = 175 in^{3}

The base area of the triangular prism = 32 in^{2}

The volume of the rectangular prism = Bh = 32 x 6 = 192 in^{3}

The triangular prism is a better deal since it has a larger volume.

**H.O.T.**

**Focus on Higher Order Thinking**

Question 16.

Critical Thinking

The wading pool shown is a trapezoidal prism with a total volume of 286 cubic feet. What is the missing dimension?

______ ft.

Answer: 3.5 ft

Explanation:

Area of the trapezoidal prism = B = 1/2 x 13 (2+x)

Volume of the figure = 286 cubic feet

V = Bh

286 = 1/2 x 13 (2+x)(8)

5.5 = (2+x)

x = 3.5 ft

Question 17.

Persevere in Problem Solving

Lynette has a metal doorstop with the dimensions shown. Each cubic centimeter of the metal in the doorstop has a mass of about 8.6 grams. Find the volume of the metal in the doorstop. Then find the mass of the doorstop.

______ grams

Answer: 75 cubic centimeter, 645 grams

Explanation:

V = Bh

B = Area of the triangle of base = 10 cm , height = 6 cm = 1/2 x 10 x 6 = 30 square centimeter

V = 30 x 2.5 = 75 cubic centimeter

1 cubic centimeter = 8.6 grams in mass

V = 75 cubic centimeter x 8.6 = 645 grams

Question 18.

Analyze Relationships

What effect would tripling all the dimensions of a triangular prism have on the volume of the prism? Explain your reasoning.

Type below:

____________

Answer: The volume is 27 times the original volume.

Explanation:

The area of the base = B = 1/2 (3b) (3h) = 9/2 (bh)

H is the height of the prism

The volume would be = 9/2 (bh) x (3H) = 27 [ 1/2 (bhH) ]

Therefore, The volume is 27 times the original volume.

Question 19.

Persevere in Problem Solving

Each of two trapezoidal prisms has a volume of 120 cubic centimetres. The prisms have no dimensions in common. Give possible dimensions for each prism.

Type below:

____________

Answer: A possible combination of dimension could be the height at 8 cm, base at 2 cm and 3 cm

Explanation:

The numbers that multiply to get 120 are 20 and 6 so let the first prism have a base area of 20 square centimetres and the height of 6 cm.

If the base area is 20, the height of the trapezoid and the length of the bases could be 8,2 and 3 respectively.

The other numbers that multiply to get 120 are 4 and 30 so let the second prism have a base area of 30 square centimetres and the height of 4 cm.

If the base area is 30, the height of the trapezoid and the length of the bases could be 10,1 and 5 respectively.

### 9.1, 9.2 Circumference and Area of Circles – Page No. 295

**Find the circumference and area of each circle. Use 3.14 for π. Round to the nearest hundredth if necessary.**

Question 1.

C = _________ m

A = _________ m^{2}

Answer:

C = 43.96 m

A = 153.86 m^{2}

Explanation:

C = 2 πr = 2 π(7) = 14 (3.14) = 43.96 m

A = πr^{2} = 3.14 (7)^{2} = 153.86 m^{2}

Question 2.

C = _________ ft

A = _________ ft^{2}

Answer:

C = 37.68 ft

A = 113.04 ft^{2}

Explanation:

Diameter = 12 ft

Radius = d/2 = 12/2 = 6 ft

C = 2 πr = 2 π(6) = 6 (3.14) = 37.68 ft

A = πr^{2} = 3.14 (6)^{2} = 113.04 ft^{2}

**9.3 Area of Composite Figures**

**Find the area of each figure. Use 3.14 for π.**

Question 3.

______ m^{2}

Answer: 180.48 m^{2}

Explanation:

Area of the triangle = 1/2 x 16 x 10 = 80 m^{2}

Area of the semicircle = 1/2 πr^{2} = 1/2 (3.14) (8)^{2} = 100.48 m^{2}

The total area of the figure = 80 + 100.48 = 180.48 m^{2}

Question 4.

______ cm^{2}

Answer: 200 cm^{2}

Explanation:

Area of the parallelogram = 4.5(20) = 90 cm^{2}

Area of the rectangle = 20(5.5) = 110 cm^{2}

The total area of the figure = 90 + 110 = 200 cm^{2}

**9.4, 9.5 Solving Surface Area and Volume Problems**

**Find the surface area and volume of each figure.**

Question 5.

S = _________ cm^{2}

V = _________ cm^{3}

Answer:

S = 132 cm^{2}

V = 60 cm^{3}

Explanation:

Perimeter = 3+4+5 = 12 cm

Base area = Area of the triangle = 1/2 x 3 x 4 = 6

S = Ph + 2B = 12(10) + 2(6) = 120 +12 = 132 cm^{2}

V = Bh = 6 x 10 = 60 cm^{3}

Question 6.

S = _________ yd^{2}

V = _________ yd^{3}

Answer:

S = 54.5 yd^{2}

V = 27.5 yd^{3}

Explanation:

Perimeter = 2(2.5) + 2(2) + 4 = 13 cm

Base area = Area of the triangle + Area of the rectangle = 1/2 x 1.5 x 4 + 4(2)= 11

S = Ph + 2B = 13(2.5) + 2(11) = 32.5 +22 = 54.5 yd^{2}

V = Bh = 11 x 2.5 = 27.5 yd^{3}

**Essential Question**

Question 7.

How can you use geometry figures to solve real-world problems?

Type below:

______________

Answer: We can solve real-world problems by finding surface area and volume.

Example: We can find the amount of liquid in a tank by calculating its volume.

Explanation:

Real-world problems by finding surface area and volume.

Example1: We can find the amount of liquid in a tank by calculating its volume.

Example2: We can find the surface area of the house and find the amount of paint required to paint the house.

### Page No. 296

Question 1.

What is the circumference of the circle?

a. 34.54 m

b. 69.08 m

c. 379.94 m

d. 1519.76 m

Answer: b. 69.08 m

Explanation:

Circumference = 2 πr = 2 π(11) = 22 (3.14) = 69.08 m

Question 2.

What is the area of the circle?

Options:

a. 23.55 m^{2}

b. 47.1 m^{2}

c. 176.625 m^{2}

d. 706.5 m^{2}

Answer: c. 176.625 m^{2}

Explanation:

Diameter = 15 m

Radius = 7.5 m

Area of the circle = πr^{2} = 3.14 (7.5)^{2} = 176.625 m^{2}

Question 3.

What is the area of the figure?

Options:

a. 28.26 m^{2}

b. 36 m^{2}

c. 64.26 m^{2}

d. 92.52 m^{2}

Answer: c. 64.26 m^{2}

Explanation:

Area of the square = 6 x 6 = 36 m^{2}

Radius = 6 m

Area of the quarter circle = 1/4 πr^{2} = 1/4 x 3.14 (6)^{2} = 28.26 m^{2}

The total area of the figure = 36 + 28.26 = 64.26 m^{2}

Question 4.

A one-year membership to a health club costs $480. This includes a $150 fee for new members that is paid when joining. Which equation represents the monthly cost x in dollars for a new member?

Options:

a. 12x + 150 = 480

b. \(\frac{x}{12}\) + 150 = 480

c. 12x + 480 = 150

d. \(\frac{x}{12}\) + 480 = 150

Answer: a. 12x + 150 = 480

Explanation:

If x is the monthly fee, then 12x is the total monthly fees.

The joining fee = $150

Total cost = $480

then,

12x + 150 = 480

Question 5.

What is the volume of the prism?

Options:

a. 192 ft^{3}

b. 48 ft^{3}

c. 69 ft^{3}

d. 96 ft^{3}

Answer: d. 96 ft^{3}

Explanation:

B = Base area of the triangle = 1/2 x 8 x 2 = 8 ft^{2}

Height = 12 ft

Volume of the triangular orism = Bh = 8(12) = 96 ft^{3}

Question 6.

A school snack bar sells a mix of granola and raisins. The mix includes 2 pounds of granola for every 3 pounds of raisins. How many pounds of granola are needed for a mix that includes 24 pounds of raisins?

Options:

a. 16 pounds

b. 36 pounds

c. 48 pounds

d. 120 pounds

e. 120 pounds

Answer: a. 16 pounds

Explanation:

2/3 is equal to x/24 then 3 times 8 is equal to 24 and if 2 times 8 is equal to 16.

Question 7.

Find the percent change from $20 to $25.

Options:

a. 25% decrease

b. 25% increase

c. 20% decrease

d. 20% increase

Answer: b. 25% increase

Explanation:

25 – 20 = 5 divide by 20 = 1/4

When we find the percentage we get 25.

So we can say that there is an increase in 25%

Question 8.

Each dimension of the smaller prism is half the corresponding dimension of the larger prism.

a. What is the surface area of the figure?

_____ in^{2}

Answer: 856 in^{2}

Explanation:

Height of the top prism = 10/2 = 5

Length of the top prism = 16/2 = 8

Width of the top prism = 8/2 = 4

Perimeter = 2l + 2w = 2(8) + 2(4) = 16 + 8 = 24 in

B = lw = 8(4) = 32 in

Surface area of top prism= Ph + 2B = 24(5) + 2(32) = 184 in^{2}

Height of the prism = 10

Length of the prism = 16

Width of the prism = 8

Perimeter = 2l + 2w = 2(16) + 2(8) = 32 + 16 = 48 in

B = lw = 16(8) = 128 in

Surface area of bottom prism= Ph + 2B = 48(10) + 2(128) = 736 in^{2}

Area of overlapping region = 32 in^{2}

The total surface area of the prism

= Surface area of top prism + Surface area of bottom prism – 2[Area of overlapping region ]

= 184 + 736 – 2(32) = 856 in^{2}

Question 8.

b. What is the volume of the figure?

_____ in^{3}

Answer: 1440 in^{3}

Explanation:

Volume of top prism = Bh = 32(5) = 160 in^{3}

Volume of bottom prism = Bh = 128(10) = 1280 in^{3}

The total volume of the figure = 160 + 1280 = 1440 in^{3}

### EXERCISES – Page No. 298

Question 1.

In the scale drawing of a park, the scale is 1 cm: 10 m. Find the area of the actual park.

_____ m^{2}

Answer: 450 m^{2}

Explanation:

Multiply the dimensions of the scale drawing by 10 since 1 cm = 10 m

3cm by 1.5 cm = 30m by 15 m

Area = 30(15) = 450 m^{2}

Question 2.

Find the value of y and the measure of ∠YPS.

y = __________ °

mYPS = __________ °

Answer: y = 8

mYPS = 40 °

Explanation:

140 + 5y = 180 [sum of angle on a line = 180°]

5y = 40

y = 8

mYPS = mRPZ = 5y [vertically opposite angles]

mYPS = 5(8) = 40°

Question 3.

Kanye wants to make a triangular flower bed using logs with the lengths shown below to form the border. Can Kanye form a triangle with the logs without cutting any of them? Explain.

_____

Answer: No

Explanation:

A side of a triangle must be greater than the difference of the other two sides and smaller than the sum of the other 2 sides.

The sum of the first 2 sides = 3+4 = 7 < 8

Therefore, he cannot form a triangle unless he cuts the logs.

Question 4.

In shop class, Adriana makes a pyramid with a 4-inch square base and a height of 6 inches. She then cuts the pyramid vertically in half as shown. What is the area of each cut surface?

_____ in^{2}

Answer: 12 in^{2}

Explanation:

Base = 4 in

Height = 6 in

Area of the triangle = 1/2 x 6 x 4 = 12 in^{2}

### Page No. 300

**Find the circumference and area of each circle. Round to the nearest hundredth.**

Question 1.

C = __________ in

A = __________ in^{2}

Answer:

C = 69.08 in

A = 379.94 in^{2}

Explanation:

Diameter = 22 in

Radius = d/2 = 22/2 = 11 in

C = 2 πr = 2 π(11) = 22 (3.14) = 69.08 in

A = πr^{2} = 3.14 (11)^{2} = 379.94 in^{2}

Question 2.

C = __________ m

A = __________ m^{2}

Answer:

C = 28.26 m

A = 63.59m^{2}

Explanation:

Radius = 4.5 m

C = 2 πr = 2 π(4.5) = 9 (3.14) = 28.26 m

A = πr^{2} = 3.14 (4.5)^{2} = 63.59 m^{2}

**Find the area of each composite figure. Round to the nearest hundredth if necessary.**

Question 3.

______ in^{2}

Answer: 99 in^{2}

Explanation:

Area of the square = 9 x 9 = 81 in^{2}

Base of the triangle = 13 – 9 = 4 in

Area of the triangle = 1/2 x 4 x 9 = 18 in^{2}

The total area of the figure = 81 + 18 = 99 in^{2}

Question 4.

______ cm^{2}

Answer: 420.48 cm^{2}

Explanation:

Area of the rectangle = 16 x 20 = 320 cm^{2}

Diameter = 16 cm

Radius = 16/2 = 8 cm

Area of the semi circle = 1/2 πr^{2} = 1/2 x 3.14 (8)^{2} = 100.48 cm^{2}

The total area of the figure = 320 + 100.48 = 420.48 cm^{2}

**Find the volume of each figure.**

Question 5.

______ in^{3}

Answer: 420 in^{3}

Explanation:

B = 7(5) = 35 in^{2}

V = Bh = 35 x 12 = 420 in^{3}

Question 6.

The volume of a triangular prism is 264 cubic feet. The area of a base of the prism is 48 square feet. Find the height of the prism.

______ in

Answer: 5.5 ft

Explanation:

V = Bh

264 = 48h

h = 264/48 = 5.5ft

### Page No. 301

**A glass paperweight has a composite shape: a square pyramid fitting exactly on top of an 8 centimeter cube. The pyramid has a height of 3 cm. Each triangular face has a height of 5 centimeters.**

Question 7.

What is the volume of the paperweight?

______ cm^{3}

Answer: 576 cm^{3}

Explanation:

Pyramid:

B = 8 x 8 = 64 cm^{2}

V = 1/3 Bh = 1/3 x 64 x 3 = 64 cm^{3}

Prism:

B = 8 x 8 = 64 cm^{2}

V = Bh = 64 x 8 = 512 cm^{3}

The total volume of the figure = 64 + 512 = 576 cm^{3}

Question 8.

What is the total surface area of the paperweight?

______ cm^{2}

Answer: 400 cm^{2}

Explanation:

Pyramid:

P = 4(8) = 32 cm

S = 1/2 Pl + B = 80 + 64 = 144 cm^{2}

Prism:

P = 4(8) = 32 cm

S = Ph + 2B = 32(8) + 2(64) = 384 cm^{2}

The total surface area of the prism

= Area of the prism + Area of the pyramid – 2[Area of the overlapping region]

= 144 + 384 – 2(64) = 400

**Unit 4 Performance Tasks**

Question 9.

Product Design Engineer

Miranda is a product design engineer working for a sporting goods company. She designs a tent in the shape of a triangular prism. The dimensions of the tent are shown in the diagram.

a. How many square feet of material does Miranda need to make the tent (including the floor)? Show your work.

______ ft^{2}

Answer: 261 3/4 ft^{2}

Explanation:

P = 2 x 7 1/2 + 8 = 22 1/2

B = 4/2 (8) (6) = 24

S = Ph + 2B = 22 1/2 x 9 1/2 + 2(24) = 213 3/4 + 48 = 261 3/4 ft^{2}

Question 9.

b. What is the volume of the tent? Show your work.

______ ft^{3}

Answer: 228 ft^{3}

Explanation:

V = Bh = 24 x 9 1/2 = 228 ft^{3}

Question 9.

c. Suppose Miranda wants to increase the volume of the tent by 10%. The specifications for the height (6 feet) and the width (8 feet) must stay the same. How can Miranda meet this new requirement? Explain

Type below:

____________

Answer: Increase the height to 10.45 ft

Explanation:

New volume = 1.10 x 228 = 250.8

250.8 = 24h

h = 10.45 ft

### Unit 4 Performance Tasks (cont’d) – Page No. 302

Question 10.

Li is making a stand to display a sculpture made in art class. The stand will be 45 centimeters wide, 25 centimeters long, and 1.2 meters high.

a. What is the volume of the stand? Write your answer in cubic centimeters.

______ cm^{3}

Answer: 135,000 cm^{3}

Explanation:

B = 45 x 25 = 1125 cm^{2}

V = Bh = 1125 x 120 = 135,000 cm^{3}

Question 10.

b. Li needs to fill the stand with sand so that it is heavy and stable. Each piece of wood is 1 centimeter thick. The boards are put together as shown in the figure, which is not drawn to scale. How many cubic centimeters of sand does she need to fill the stand? Explain how you found your answer.

______ cm^{3}

Answer: 116,702 cm^{3}

Explanation:

Width = 45 – 2(1) = 43 ft

Length = 25 – 2(1) =23ft

Height = 120-2(1) = 118ft

B = 43 x 23 = 989 ft^{2}

V = Bh = 989 x 118 = 116,702 ft^{3}

### Selected Response – Page No. 303

Question 1.

A school flag is in the shape of a rectangle with a triangle removed as shown.

What is the measure of angle x?

Options:

a. 50°

b. 80°

c. 90°

d. 100°

Answer: d. 100°

Explanation:

x = 50 + 50 = 100° [ Sum of two angles created by the 2 lines]

Question 2.

On a map with a scale of 2 cm = 1 km, the distance from Beau’s house to the beach is 4.6 centimetres. What is the actual distance?

Options:

a. 2.3 km

b. 4.6 km

c. 6.5 km

d. 9.2 km

Answer: a. 2.3 km

Explanation:

2/1 = 4.6/x

x = 4.6/2 = 2.3 km

Question 3.

Lalasa and Yasmin are designing a triangular banner to hang in the school gymnasium. They first draw the design on paper. The triangle has a base of 5 inches and a height of 7 inches. If 1 inch on the drawing is equivalent to 1.5 feet on the actual banner, what will the area of the actual banner be?

Options:

a. 17.5 ft^{2}

b. 52.5 ft^{2}

c. 39.375 ft^{2}

d. 78.75 ft^{2}

Answer: c. 39.375 ft^{2}

Explanation:

1in = 1.5ft

The base of the triangle = 5 in = 1.5(5) ft = 7.5 ft

Height = 7 in = 7(1.5) ft = 10.5 ft

Area of the triangle = 1/2 x 7.5 x 10.5 = 39.375 ft^{2}

Question 4.

Sonya has four straws of different lengths: 2 cm, 8 cm, 14 cm, and 16 cm. How many triangles can she make using the straws?

Options:

a. no triangle

b. one triangle

c. two triangles

d. more than two triangles

Answer: b. one triangle

Explanation:

The third side of a triangle must be smaller than the sum of the other two sides to form a triangle.

2+8 = 10<14

2+8 = 10<16

8+14 = 22>14

8+14 = 22>16

2+14 = 16=16

2+16 = 18>16

Therefore, only one triangle can be formed using the sides 8, 14, 16.

Question 5.

A one-topping pizza costs $15.00. Each additional topping costs $1.25. Let x be the number of additional toppings. You have $20 to spend. Which equation can you solve to find the number of additional toppings you can get on your pizza?

Options:

a. 15x + 1.25 = 20

b. 1.25x + 15 = 20

c. 15x − 1.25 = 20

d. 1.25x − 15 = 20

Answer: b. 1.25x + 15 = 20

Explanation:

If x is the number of additional toppings, then 1.25 x is the cost of the additional toppings.

This gives the total cost is 1.25x + 15

then,

1.25x + 15 = 20

Question 6.

A bank offers a home improvement loan with simple interest at an annual rate of 12%. J.T. borrows $14,000 over a period of 3 years. How much will he pay back altogether?

Options:

a. $15680

b. $17360

c. $19040

d. $20720

Answer: c. $19040

Explanation:

Simple interest = 14,000 x 0.12 x 2 = $5,040

Amount = $14,000 + $5,040 = $19040

Question 7.

What is the volume of a triangular prism that is 75 centimeters long and that has a base with an area of 30 square centimeters?

Options:

a. 2.5 cm^{3}

b. 750 cm^{3}

c. 1125 cm^{3}

d. 2250 cm^{3}

Answer: d. 2250 cm^{3}

Explanation:

V = Bh = 30(75) = 2250cm^{3}

Question 8.

Consider the right circular cone shown.

If a vertical plane slices through the cone to create two identical half cones, what is the shape of the cross section?

Options:

a. a rectangle

b. a square

c. a triangle

d. a circle

Answer: c. a triangle

Explanation:

Slicing through the vertex to create 2 identical half cones would create a cross-section that is a triangle.

### Page No. 304

Question 9.

The radius of the circle is given in meters. What is the circumference of the circle? Use 3.14 for π.

a. 25.12 m

b. 50.24 m

c. 200.96 m

d. 803.84 m

Answer: b. 50.24 m

Explanation:

Circumference = 2 πr = 2 π(8) = 16 (3.14) = 50.24 m

Question 10.

The dimensions of the figure are given in millimeters. What is the area of the two-dimensional figure?

Options:

a. 39 mm^{2}

b. 169 mm^{2}

c. 208 mm^{2}

d. 247 mm^{2}

Answer: c. 208 mm^{2}

Explanation:

Area of the square = 13 x 13 = 169 mm^{2}

Area of the triangle = 1/2 x 13 x 6 = 39 mm^{2}

The total area of the figure = 169 + 39 = 208 mm^{2}

Question 11.

A forest ranger wants to determine the radius of the trunk of a tree. She measures the circumference to be 8.6 feet. What is the trunk’s radius to the nearest tenth of a foot?

Options:

a. 1.4 ft

b. 2.7 ft

c. 4.3 ft

d. 17.2 ft

Answer: a. 1.4 ft

Explanation:

Circumference = 2 πr = 8.6 ft

r = 8.6/2 π = 1.4 ft

Question 12.

What is the measure in degrees of an angle that is supplementary to a 74° angle?

Options:

a. 16°

b. 74°

c. 90°

d. 106°

Answer: d. 106°

Explanation:

Sum of supplementary angles = 180°

x + 74° = 180°

x = 106°

Question 13.

What is the volume in cubic centimeters of a rectangular prism that has a length of 6.2 centimeters, a width of 3.5 centimeters, and a height of 10 centimeters?

Options:

a. 19.7 cm^{3}

b. 108.5 cm^{3}

c. 217 cm^{3}

d. 237.4 cm^{3}

Answer: c. 217 cm^{3}

Explanation:

V = Bh

B = 6.2 x 3.5 = 21.7 cm^{2}

h = 10 cm

V = 21.7 x 10 = 217 cm^{3}

Question 14.

A patio is the shape of a circle with diameter shown.

What is the area of the patio? Use 3.14 for π.

Options:

a. 9 m^{2}

b. 28.26 m^{2}

c. 254.34 m^{2}

d. 1017.36 m^{2}

Answer: c. 254.34 m^{2}

Explanation:

Diameter = 18 m

Radius = 18/2 = 9 m

Area of the patio = πr^{2} = 3.14 (9)^{2} = 254.34 m^{2}

Question 15.

Petra fills a small cardboard box with sand. The dimensions of the box are 3 inches by 4 inches by 2 inches.

a. What is the volume of the box?

______ in^{3}

Answer: 24 in^{3}

Explanation:

V = Bh

B = 3 x 4 = 12 in^{2}

V = 12 x 2 = 24 in^{3}

Question 15.

b. Petra decides to cover the box by gluing on wrapping paper. How much wrapping paper does she need to cover all six sides of the box?

______ in^{2}

Answer: 76 in^{2}

Explanation:

P = 2(3) + 2(4) = 6 + 8 = 14 in

S = Ph + 2B = 14 x 2 + 2 x 24 = 76 in^{2}

Question 15.

c. Petra has a second, larger box that is 6 inches by 8 inches by 4 inches. How many times larger is the volume of this second box? The surface area?

Volume is _________ times greater.

Surface area is _________ times greater

Answer: Surface area is about 2.7 times larger

Explanation:

B = 6 x 8 = 48 in^{2}

V = Bh = 48 x 4 = 192 in^{3}

192/24 = 8

P = 2(6) + 2(8) = 12 + 16 = 28

S = Ph + 2B = 28(4) + 2(48) = 112 + 96 = 208 in^{2}

208/76 = 2.7