Nc Math 3.Indd

CHAPTER 9 Geometric Modeling

9.1 Modeling with Area ...... 341 9.2 Modeling with Volume ...... 347 9.3 Cross Sections of Solids ...... 355 9.4 Solids of Revolution ...... 361

Chapter Maintaining Mathematical Proficiency 9 Find the area of the circle or regular polygon. 1. 2.

3.7 ft 4.7 in. 5 ft

3. a circle with a diameter of 74.6 centimeters

4. a regular hexagon with a perimeter of 42 yards and an apothem of 4.25 yards

5. a circle with a circumference of 24π meters

Find the surface area and volume of the solid.

6. 7. 35 cm

8 in.

6 in. 28 cm 12 in.

8. 9. 37 m 58 yd

23 m

Modeling with Area 9.1 For use with Exploration 9.1

Essential Question How can you use the population and area of a region to describe how densely the region is populated?

1 EXPLORATION: Exploring Population and Area

Work with a partner. Use the Internet to find the population and land area of each county in California. Then find the number of people per square mile for each county.

a. Mendocino County b. Lake County c. Yolo County

d. Napa County e. Sonoma County f. Marin County

9.1 Modeling with Area (continued)

2 EXPLORATION: Analyzing Population and Area

Work with a partner. The six counties in Exploration 1 appear on a map as shown.

a. Without calculating, how would you expect the number of people per square mile in the entire 6-county region to compare to the values for each individual county in Exploration 1?

b. Use the populations and land areas in Exploration 1 to justify your answer in part (a).

Communicate Your Answer

3. How can you use the population and area of a region to describe how densely the region is populated?

4. Find the population and land area of the county in which you live. How densely populated is your county compared to the counties in Exploration 1?

5. In Exploration 1, the two northern counties are less densely populated than the other four. What factors do you think might influence how densely a region is populated?

Notetaking with Vocabulary 9.1 For use after Lesson 9.1

In your own words, write the meaning of each vocabulary term. Name ______Date ______population density

Practice 9.1 For use after Lesson 9.1

In your own words, write the meaning of each vocabulary term. Notes: population density

Worked-Out Examples number of people Chapter 1 Population density = —— area of land Notes:Example #1 210, 000 = — 452.39 Step 3 Sketch a diagram of the corral that includes the gate Find the indicated measure. opening, as shown. 4. The area of the region is About 650, 000 people≈ 464.20 live in a circular region with a 6-mile radius. Find the Thepopulation 2population density density in people is about per 464 square people mile. per square mile. barn 5. The area of the region is π62 = 36π ≈ 113.1 square miles. 20 m corral The population density is: number of people 650, 000 Population density = —— = — area of land 113.1 37 m 3 m ≈ 5747.1 The population density is about 5747 people per square mile. So, you need 2w + ℓ − 3 = 2(20) + 40 − 3 = 77 6. 12. The orange island has numbera greater of population animals density. If the meters of fencing. = —— area of land Example #2 4. islandsame number has a smaller of people area, live 445then on there both areislands, more and people the orangeper = — ≈ 0.0002 square mile. 2, 220, 000 Chapter 1 Smaller bearing Larger bearing MODELING WITH MATHEMATICS A soccer field of length ℓ and width w has a perimeter of 320 yards.

d = 12 mm d = 18 mm Dimensions 12 18 a. Write an expression that represents the area of the scoccer field in terms of ℓ. 8. Find the area of Central r = — Park. = 6 mmBecause it ris = approximately — = 9 mm 2 2 shaped like a rectangle, use the formula for the area of a b. Use your expression from part (a) to determine the dimensions of the field that 2 2 rectangle. S = 4πr S = 4πr maximize the area. What do you notice? 4 (6)2 4 (9)2 Surface area = π 2 = π A = ℓw = 2.5(0.5) = 1.25 mi 2 2 = 144π mm 1 = 324π mm So, the area of Central Park is 1.252 ÷ — = 1.25(640) 2= 13. Thea. Use population what you density know aboutis about the 0.0002 area and grizzly perimeter bear perof the acre. ≈ 452.39 mm ( 640 ≈) 1017.88 mm 800 acres. Elk:soccer fi eld to fi nd an expression that represents the area of the fi eld. number of animals Use the formula for population density. Let324 pπ represent9 3the2 Population density = —— The surface area of the larger bearing is — = — = — The perimeter P of a fi eldarea of of length land ℓ and width w is number of people in the park that afternoon.144π 4 ( 2) P = 2ℓ + 2w. So, 32020, = 000 2ℓ + 2w. times the surface area numberof the smaller of people bearing. = — ≈ 0.009 Population density = —— 2, 220, 000 area of3 land2 9 320 = 2ℓ + 2w no; The larger bearing needsp — = — times the amount of The population density is about 0.009 elk per acre. 15 = — (2) 4 320 − 2ℓ = 2w grease to coat it. 800 Mule deer: 320 − 2ℓ 2w — = — number of animals 1.1 Exercises (pp.15(800) 7–8) = p Population density2 = —— 2 area of land 12, 000 = p 160 − ℓ =w Copyright © Big Ideas Learning, LLC Vocabulary and Core Concept Check 2400 337 All rights reserved. So, there are about 12,000 people in Central Park that Solving for w gives= —w = 160 − ≈ℓ .0.001 1. Sample answer: The population of a state is the total number 2, 220, 000 afternoon. of people who live in that state. The population density of a The So, population an exp ression density that is represents about 0.001 the mule area deerof the per soccer acre. fi eld is A = ℓw = ℓ(160 −ℓ). 9. Usestate the is theformula number for of population people per density. square Let mile r representin that state. the Bighorn sheep: Use the table feature of a graphing calculator to create radius of the region. b. number of animals 2. The expression that doesnumber not belong of people with the other three is Populationa table of density values =to —— fi nd the length ℓ Copyright © Big Ideas Learning, LLC Population density = —— area of land “video gamers per city.” Aarea city of is land not a measurement of area. value of (160 ). 343 All rights reserved. ℓ − ℓ 260 79, 000 = — ≈ 0.0001 XY1 2, 220, 000 513 = — 2 Monitoring Progress andπ Modelingr with Mathematics 65 6175 The 70population6300 density is about 0.0001 bighorn sheep per 3. Step 1 Find the area2 of Kansas. It is approximately shaped 75 6375 513πr = 79, 000 80 6400 acre.85 6375 like a rectangle. Use the formula for the area of a 90 6300 2 79, 000 95 6175 rectangler to = estimate — the area of Kansas. X=80 513π 7. Use the formula for population density. Let p represent the 2 A = ℓw = 400(210)— = 84, 000 mi population The length of thethat region. maximizes the value of ℓ(160 − ℓ) 79, 000 Step 2 Find the populationr = — density. is 80 yards. So, the width of the fi eld is √ 513 number of people π Populationw 160 density =160 —— 80 80 yards. Because the width number of people = −ℓ = −area of= land Populationr density≈ 7 = —— and the length are both 80 yards, the area of the fi eld is area of land p maximized6366 when = it —is a square.2 The radius of the circular region2, 850, is about 000 7 miles. π(4) = — ≈ 34 84, 000 6366(16π) = p 10. Use the formula for population density. Let r represent the 14. a. Find the lateral area of the cylindrical roller, including the radius of The the population region. density is about 34 people per square 101,856π =1 p 5 1 13 13 nap. The radius is — 1 — = — — = — . mile. number of people 319,990.1 2p⋅ 8 2 ⋅ 8 16 Population density = —— ≈ area of land L = 2πrh There are about 319,990 people who live in the region. 1,150,000 13 1 18, 075 = — L = 2π — + — 9 πr2 ⋅ (16 4 ) ⋅ 18,075 r2 1,150,000 17 π = L = 2π — 9 ⋅ ( 16) ⋅ Copyright © Big Ideas Learning,1,150,000 LLC Integrated Mathematics III 3 r2 = — 153 All rights reserved. 18,075π L = — π ≈ 60.1Worked-Out Solutions — 8 ⋅ 1,150,000 So, the roller covers about 60.1 square inches in one r = — √ 18,075π revolution. r ≈ 4.5 b. Find the circumference of the cylindrical roller, including the nap. The radius of the circular region is about 4.5 kilometers. C = 2πr 11. The length of the radius should be used, not the length of 13 1 C = 2π — + — the diameter. The radius of the region is ⋅ ( 16 4) 7.5 ÷ 2 = 3.75 miles. 17 x C = 2π — 1550 = — ⋅ (16) π 3.752 ⋅ 17 x C = — π ≈ 6.7 1550 = — 14.0625 8 ⋅ π So, in one revolution, you are about 6.7 inches up the 68, 477 ≈ x wall, and in 8 full revolutions, you are about So, about 68,477 people live in the region. 8(6.7) = 53.6 inches, or about 53.6 ÷ 12 ≈ 4.5 feet up the wall. So, you are halfway up the wall. 4 Integrated Mathematics III Copyright © Big Ideas Learning, LLC Worked-Out Solutions All rights reserved. Chapter 1

8. Find the area of Central Park. Because it is approximately 12. The orange island has a greater population density. If the shaped like a rectangle, use the formula for the area of a same number of people live on both islands, and the orange rectangle. island has a smaller area, then there are more people per A = ℓw = 2.5(0.5) = 1.25 mi2 square mile. 1 So, the area of Central Park is 1.25 ÷ — = 1.25(640) = 13. a. Use what you know about the area and perimeter of the ( 640) 800 acres. soccer fi eld to fi nd an expression that represents the area of the fi eld. Use the formula for population density. Let p represent the number of people in the park that afternoon. The perimeter P of a fi eld of lengthℓ and width w is number of people P = 2ℓ + 2w. So, 320 = 2ℓ + 2w. Population density = —— area of land 320 = 2ℓ + 2w p 15 = — 320 − 2ℓ = 2w 800 320 − 2ℓ 2w — = — 15(800) = p 2 2 12, 000 = p Name 160 ______− ℓ =w Date ______So, there are about 12,000 people in Central Park that Solving for w gives w = 160 −ℓ. afternoon. 9.1 Practice (continued) fi eld = is A = ℓ w ℓ −(160 ℓ). 9. Use the formula for population density. Let r represent the radius of the region. ExtraPracticeb. Use the Practice table A feature of a graphing calculator to create number of people a table of values to fi nd the length ℓ that maximizes the Population density = —— 1. About 70,000 people live in a circular region with a 30-mile radius. Find the population area of land value of ℓ(160 − ℓ). 79, 000 density in people per square mile. XY1 513 = — 2 πr 65 6175 70 6300 2 75 6375 513πr = 79, 000 80 6400 85 6375 90 6300 2 79, 000 95 6175 r = — X=80 513π 2. About 370,000 people live in a circular region with a 5-mile radius. Find the population — The length that maximizes the value of ℓ(160 − ℓ) 79, 000 density in people per square mile. r = — is 80 yards. So, the width of the fi eld is √ 513π Namew = ______160 − ℓ = 160 − 80 = 80 yards. Because the width Date ______Name ______Date ______r ≈ 7 and the length are both 80 yards, the area of the fi eld is The radius of the circular region is about 7 miles. 9.1maximized Notetakin when it isg a withsquare. Vocabulary (continued) 9.1 Notetaking with Vocabulary (continued) 14. a. Find the lateral area of the cylindrical roller, including the 10. Use the formula for population density. Let r represent the Extra Practice3. A mapPractice of A the state of Montana is approximately rectangular with a length of 590 miles radius of the region. and a width of 2501 miles.5 1 13 13 ExtraPractice nap. The Practice radius A is — 1 — = — — = — . number of people About 70,000 people2 ⋅ 8 live 2in⋅ a circular8 16 region with a 30-mile radius. Find the population Population density = —— 1. area of land 1.L = Aboutdensitya. 2 π Montanarh 70,000 in people has people aper population squarelive in amile. ofcircular about region990,000. with Find a 30-mile the population radius. Finddensity the in population people 1,150,000 densityper squarein13 people 1mile. per square mile. 18, 075 = — L = 2π — + — 9 πr2 ⋅ (16 4 ) ⋅

18,075πr2 = 1,150,000 17 L = 2π — 9 ⋅ (16) ⋅ 2 1,150,000 r = — 2. About153 370,000 people live in a circular region with a 5-mile radius. Find the population 18,075π L = — π ≈ 60.1 b. About The 370,000 table shows people the live estimated in a circular populations region forwith domestic a 5-mile sheep radius. and Find cattle the inpopulation Montana. — 2. density8 ⋅ in people per square mile. 1,150,000 So, densitythe Findroller in the coverspeople population aboutper square 60.1density squaremile. in animals inches inper one square mile for each animal. r = — √ 18,075π revolution. Find the circumference of the cylindrical roller, including r ≈ 4.5 b. Animal Sheep Cattle the nap. The radius of the circular region is about 4.5 kilometers. Population 225,000 2.6 million C = 2πr 3. A map of the state of Montana is approximately rectangular with a length of 590 miles 11. The length of the radius should be used, not the length of 3. A map of13 the state1 of Montana is approximately rectangular with a length of 590 miles C =and 2π a width — +of —250 miles. the diameter. The radius of the region is and a⋅ width(16 of 4250) miles. 7.5 ÷ 2 = 3.75 miles. a. Montana17 has a population of about 990,000. Find the population density in people x C =a. 2 π Montana — has a population of about 990,000. Find the population density in people 1550 = — per⋅ (square16) mile. π 3.752 ⋅ 17per square mile. x 4.C = About — 860,000π ≈ 6.7 people live in a circular region with a population density of about 6480 people 1550 = — 14.0625 8 ⋅ π So, perin one square revolution, mile. Find you theare radiusabout of6.7 the inches region. up the 68, 477 ≈ x wall, and in 8 full revolutions, you are about

So, about 68,477 people live in the region. 8(6.7) = 53.6 inches, or about 53.6 ÷ 12 ≈ 4.5 feet up b. The table shows the estimated populations for domestic sheep and cattle in Montana. the wall. So, you are halfway up the wall. b. TheFind table the population shows the densityestimated in animalspopulations per squarefor domestic mile for sheep each and animal. cattle in Montana. 4 Integrated Mathematics III Find the populationCopyright density © Big in animals Ideas Learning, per square LLC mile for each animal. Worked-Out Solutions All rights reserved. Animal Sheep Cattle Animal Sheep Cattle Population 225,000 2.6 million Population 225,000 2.6 million

Copyright © Big Ideas Learning, LLC All rights reserved. 344 4. 4. Aboutper square 860,000 mile. people Find thelive radius in a circular of the region.region with a population density of about 6480 people per square mile. Find the radius of the region.

9.1 Notetaking with Vocabulary (continued)

ExtraPractice Practice A 1. About 70,000 people live in a circular region with a 30-mile radius. Find the population density in people per square mile.

2. About 370,000 people live in a circular region with a 5-mile radius. Find the population density in people per square mile.

3. A map of the state of Montana is approximately rectangular with a length of 590 miles and a width of 250 miles.

a. Montana has a population of about 990,000. Find the population density in people per square mile.

b. The table shows the estimated populations for domestic sheep and cattle in Montana. Find the population density in animals per square mile for each animal.

Animal Sheep Cattle Population 225,000 2.6 million

Name ______Date ______

9.1 Practice (continued)

4.5. AboutYou have 860,000 350 yards people of livefencing in a tocircular build aregion rectangular with a corralpopulation of length density A and of aboutwidth 6480w. people per square mile. Find the radius of the region. Namea. ______Write an expression that represents the area of the corral in terms of A. Date ______

9.1 Notetaking with Vocabulary (continued)

b. Use your expression from part (a) to determine the dimensions of the corral that You have 350 yards of fencing to build a rectangular corral of length A and width w. 5. maximize the area. a. Write an expression that represents the area of the corral in terms of A.

b. Use your expression from part (a) to determine the dimensions of the corral that In Exercises 6 and 7, describe how the change affects the surface area of the right maximize the area. prism or cylinder. Copyright © Big Ideas Learning, LLC All rights6. doublingreserved. all linear dimensions 7. multiplying338 the height by 4

5 ft In Exercises 6 and 7, describe how the change affects the surface area of the right prism or cylinder. 8 ft 5 ft 6. doubling all linear dimensions 7. multiplying the height by 4 13 ft 3 ft

5 ft

8 ft 8. You are wrapping a birthday present that is a rectangular prism. The present is 265 inches ft long, 13 ft 12 inches tall,3 ft and 20 inches wide.

a. What is the minimum area of wrapping paper required to cover the box?

8. You are wrapping a birthday present that is a rectangular prism. The present is 26 inches long, b. What is the minimum area of wrapping paper required if you place an identical box 12 inches tall, and 20 inches wide. on top of the original and wrap them together? What is the minimum area of wrapping paper required to cover the box? a.

c. Should you cut your wrapping paper to the minimum area you found in parts (a) and (b)? b. WhatExplain. is the minimum area of wrapping paper required if you place an identical box on top of the original and wrap them together?

c. Should you cut your wrapping paper to the minimum area you found in parts (a) and (b)? Explain.

Practice1.1 BPractice B

In Exercises 1–4, find the indicated measure. 1. A state park has an area of 112 acres. The table shows the estimated park populations for several animals. Find the population density in animals per acre for each animal.

Animal Otter Raccoon Fox Bobcat Population 35 186 9 3

2. A city park is triangular with a base length of 4 blocks and a height of 7 blocks. During an evening concert, its population density is about 54 people per square block. Find the number of people in the park that evening.

3. About 150,000 people live in a circular region with a population density of about 1578 people per square mile. Find the radius of the region.

4. About 1.75 million people live in a circular region with a population density of about 5050 people per square mile. Find the radius of the region.

In Exercises 5 and 6, describe how the change affects the surface area of the right prism or right cylinder. 5. doubling the diameter 6. multiplying the base edge by 2 and the height by 1 3 cm 3

5 cm 15 yd

8 yd

7. A baseball with a 2.9-inch diameter has a layer of leather on its surface.

4 a. Does a softball with a diameter that is times the diameter of the 3 baseball need 4 times the amount of leather? Explain. 3

b. What is the radius of a softball that uses four-thirds of the amount of leather used to cover the 2.9-inch baseball?