This Packet Belongs to Student Name Topic 8

This Packet Belongs to ____________ (Student Name) Topic 8: Measurement and Modeling in Two and Three Dimensions Unit 7– Measurement and Modeling in Two and Three Dimensions Module 18: Volume Formulas 18. 1 18. 2 18. 3 18. 4 Volume of Prisms and Cylinders Volume of Pyramids Volumes of Cones Volume of Spheres Module 19: Visualizing Solids 19. 1 Cross-Sections and Solids of Rotation 19. 2 Surface Area of Prisms and Cylinders 19. 3 Surface Area of Pyramids and Cones 19. 4 Surface Area of Spheres Module 20: Modeling and Problem Solving 20. 1 Scale Factors 20. 2 Modeling and Density 20. 3 Problem Solving with Constraints 1

Quick Access to Module Lesson. Click on desired Lesson 2

VOLUME FORMULAS Objectives Use formulas routinely for finding the perimeter and area of basic prisms, pyramids, cylinders, cones, and spheres. Vocabulary: Volume, right vs oblique Assignments: 3

10 -2 Volume of Prisms and Cylinders Warm Up Find the area of each figure described. Use 3. 14 for . 1. a triangle with a base of 6 feet and a height of 3 feet 9 ft 2 2. a circle with radius 5 in. 78. 5 in 2 3. 36 mm 2 Holt CA Course 1 4

Volume • The volume of a three-dimensional figure is the number of cubes it can hold. Each cube represents a unit of measure called a cubic unit. 5

Prisms and Cylinders and its Parts Rectangular prism Triangular prism Height Base Cylinder Base Remember! Knowing the SHAPE & area of the BASE is very important. They do not give you base on ref. sheet. 6

Pyramids, Cones TIP These have a POINT 7

Sphere TIP If they give you the diameter, find the RADIUS by diving by 2 8

Guess the m&m’s in a Cylinder 9

General Formula for Prims • AREAS of BASES If BASE is Rectangle/Square If BASE is Triangle If BASE is Circle What if they give you diameter? 10

If the shape has a POINT • AREAS of BASES If BASE is Rectangle/Square If BASE is Triangle If BASE is Circle 11

Volume of a Sphere TIP! Plug in 3. 14 for PI 12

Class Activity • Identify the NAME of the figure • • • Prism (triangular or rectangular) Cylinder Pyramid ((triangular or rectangular) Sphere Cone • Give the Formula of the shape. 13

Cone 14

Rectangular Pyramid 15

Sphere 16

Triangular Prism 17

Cone 18

Triangular Pyramid 19

Triangular Prism

Rectangular Prism 21

Rectangular Pyramid 22

Cone 23

Rectangular Pyramid 24

Sphere 25

Warm Up Multiply. Round to nearest hundedth 1. (2 ft)(3. 14)(5 ft)(10 ft)? 2. (5. 5 in)(1/2 in)(3. 14) 314 ft 3 8. 64 in 2 26

• Identify the shape 1. Identify • Find the area of the base 2. Area Base Steps to find the Volume • Multiply by the height 3. Height • If it has a point, ALSO multiply by 1/3 Point? Sphere? 27

Example 1 A: Finding the Volume of Prisms Find the volume of each figure to the nearest tenth. A. 1. Identify Shape Rectangular Prism Does it have a point? V = Bh B = 4 • 12 = 48 ft 2 2. Find Area of Base What is the Base? A rectangle V= 48 • 4 3. Multiply by Height V= 192 ft 3 4. Does it have a point? No. SO you’re done. 28

Example 1 B: Finding the Volume of a Cylinders Find the volume of the figure to the nearest tenth. Use 3. 14 for . B. 1. Identify Shape 2. Find Area of Base Cylinder What is the Base? A circle Does it have a point? B = (r 2) V = Bh B = (42) = 16 in 2 3. Multiply by Height = 16 • 12 in 4. Does it have a point? No. SO you’re done. = 192 602. 9 in 3 29

Example 1 C: Finding the Volume of Prisms Find the volume of the figure to the nearest tenth. Use 3. 14 for . 1. Identify Shape Triangular prism 2. Find Area of Base Does it have a point? V = Bh What is the Base? A triangle 3. Multiply by Height V = 15 • 7 7 ft 4. Does it have a point? No. SO you’re done. V= 105 ft 3 30

Example 3 C: Finding Volumes of Cones Find the volume of a cone. Step 1 Use the Pythagorean Theorem to find the height. 162 + h 2 = 342 h 2 = 900 h = 30 Pythagorean Theorem Subtract 162 from both sides. Take the square root of both sides. Step 2 Use the radius and height to find the volume. Volume of a cone Substitute 16 for r and 30 for h. 2560 cm 3 8042. 5 cm 3 Simplify. 31

Example 1 D: Finding the Volume of Pyramids Find the volume of the figure to the nearest tenth. Use 3. 14 for . 1. Identify Shape Rectangular Pyramid Does it have a point? 2. Find Area of Base What is the Base? A rectangle 3. Multiply by Height = 64 • 9 4. Does it have a point? 32

Example 1 E: Finding the Volume of Pyramids A sphere is the locus of points in space that are a fixed distance from a given point called the center of a sphere. A radius of a sphere connects the center of the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres 33

1 E. Sphere– Volume Find the volume of the sphere. Give your answer in terms of . Volume of a sphere. = 2304 in 3 Simplify. 34

2 E. Sphere– Diameter via Volume Find the diameter of a sphere with volume 36, 000 cm 3. Volume of a sphere. Substitute 36, 000 for V. 27, 000 = r 3 r = 30 d = 60 cm Take the cube root of both sides. d = 2 r 35

3 E. Sphere– Volume Find the volume of the hemisphere. Volume of a hemisphere Substitute 15 for r. = 2250 m 3 Simplify. 36

YOUR TURN Find the volume of the figure to the nearest tenth. Use 3. 14 for . A. 10 in. 6 in. 3 in. B = 6 • 3 = 18 in. 2 The base is a rectangle. V = Bh Volume of prism = 18 • 10 Substitute for B and h. = 180 in 3 Multiply. 37

YOUR TURN Find the volume of the figure to the nearest tenth. Use 3. 14 for . B. B = (82) The base is a circle. = 64 cm 2 8 cm Volume of a V = Bh cylinder 15 cm Substitute for B = (64 )(15) = 960 and h. 3, 014. 4 cm 3 Multiply. 38

YOUR TURN Find the volume of the figure to the nearest tenth. C. 10 ft The base is a triangle. = 60 ft 2 V = Bh 14 ft = 60(14) 12 ft = 840 ft 3 Volume of a prism Substitute for B and h. Multiply. 39

Your Turn! Example E Find the Surface Area of the sphere. 2. Find the radius of a sphere with volume 2304 ft 3. Volume of a sphere Substitute for V. r = 12 ft Simplify. 40

Round to HONORS--Complete this table Solid Formula D = 10 cm L = 15 cm H = 13 cm R = 3. 5 cm L = 9 cm H = 60 m L = 50 m W = 40 m D = 23 mm Volume 41

Additional Example 2: Music Application A drum company advertises a snare drum that is 4 inches high and 12 inches in diameter. Estimate the volume of the drum. 1. Identify Shape & Parts 3. Multiply by Height Cylinder V = Bh d = 12, h = 4 r=6 2. Find Area of Base What is the Base? Circle B = r 2 B = (6)2 (use 3. 14 for pi) 4. Does it have a point? No. SO you’re done. The volume of the drum is approximately 452 in 3. 42

YOUR TURN A drum company advertises a bass drum that is 12 inches high and 28 inches in diameter. Estimate the volume of the drum. d = 28, h = 12 r = 14 B = ( r 2) BASE is area of circle V = ( r 2)h = (3. 14)(14)2 • 12 Multiply by height Use 3. 14 for . = (3. 14)(196)(12) = 7385. 28 ≈ 7, 385 The volume of the drum is approximately 7, 385 in 3. 43

What is a Cross Section? A Circle A triangle A rectangle 44

Cavalieri’s Principle • Cavalieri’s Principle essentially states that if two prisms have the property that all corresponding cross sections have the same area, then those prisms have the same volume. A right prism and an oblique prism with the same base and height have the same volume. REMEBER Oblique means “slanted” The CDs stacks have the same Volume 45

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Example 3: Finding Volumes of Oblique Cylinders Find the volume of the cylinder. Give your answers in terms of and rounded to the nearest tenth. V = Bh Volume of a cylinder V = r 2 h Base is a circle. What is area of Circle? = (9)2(14) = 1134 in 3 3562. 6 in 3 47

YOUR TURN Directions: Find the volume. 48

Volume Formulas Prisms and Cylinders Cube: V = x 3 Rectangular Prism: V = lwh or V = Bh Triangular Prism: V = Bh Cylinder: V = � r 2 h or V = Bh **Basically, V = Bh for any Prism or Cylinder 49

Area of Composite figures Objectives Apply and combine the volume formulas routinely composite figures Vocabulary: Composite figures Assignments: 50

Composite Figures • A composite figure is a figure that is made up of two or more geometric figures It could be two volumes ADDED It could be two volumes SUBTRACTED OR 51

Example 4 A: Finding the Volume of Composite Figures Find the volume of the barn. Volume of barn = Volume of rectangular prism + Volume of triangular prism = 30, 000 + 10, 000 = 40, 000 ft 3 The volume of the barn is 40, 000 ft 3. 52

Example 4 B: Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. The volume of the rectangular prism is: V = ℓwh = (8)(4)(5) = 160 cm 3 The base area of the regular triangular prism is: The volume of the regular triangular prism is: An equilateral Triangle? Think 30 -60 -90 special Right triangle The total volume of the figure is the sum of the volumes. 53

YOUR TURN Find the volume of the play house. Volume of house = Volume of rectangular prism + Volume of triangular prism = 96 + 60 5 ft 4 ft 8 ft 3 ft V = 156 ft 3 The volume of the play house is 156 ft 3. 54

Composite Figure a cone in a cone or what not. 55

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Example 5 Find the volume of the composite figure. Round to the nearest tenth. Find the side length s of the base: The volume of the square prism is: The volume of the cylinder is: The volume of the composite is the cylinder minus the rectangular prism. Vcylinder — Vsquare prism = 45 — 90 51. 4 cm 3 This means “volume of Cylinder” 57

Honors LESSON QUIZ COMING SOON 58

Classwork 59

Cross Sections & 3 d figures Objectives Identify cross sections of 3 D figures Draw the Vocabulary: • Cross section Assignments: 60

What is a Cross Section? A Circle A triangle A rectangle 61

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Surface area Objectives Learn and apply the formula for the surface area of a prism, cylinder, pyramid, cones, and spheres Vocabulary: Base, lateral faces, perimeter Assignments: 66

Surface Area of Prisms & Cylinders. 67

Surface Area • The surface area of each of these solids is the sum of the areas of all the faces or surfaces that enclose the solid. For prisms and pyramids, the faces include the solid’s bases and its lateral faces. • How will the answer be labeled? • Units 2 because it is area! 68

Let’s start in the beginning… Before you can do surface area or volume, you have to know the following formulas. Rectangle Triangle Circle A = lw A = ½ bh A = π r² Steps for Finding Surface Area 1. Draw and label each face of the solid as if you had cut the solid apart along its edges and laid it flat. Label the dimensions. 2. Calculate the area of each face. If some faces are identical, you only need to find the area of one. 3. Find the total area of all the faces 69

Warm Up Find the perimeter and area of each polygon. 1. a rectangle with base 14 cm and height 9 cm P = 46 cm; A = 126 cm 2 2. a right triangle with 9 cm and 12 cm legs. P = 36 cm; A = 54 cm 2 3. an equilateral triangle with side length 6 cm 70

What is a Geometric Net? http: //www. mathsisfun. com/definitions/net. html 71

A. Triangular Prism – Surface Area 1. Draw and label the two congruent bases, and the four lateral faces unfolded into one rectangle. Then find the areas of all the rectangular faces. 2. Find individual areas and add 72

B. Triangular Prism – Surface Area How many triangles? 73

C. Cube– Surface Area BACK SIDE BOTTO M SIDE FRONT TOP 74

D. Cylinder– Surface Area Find the lateral area and surface area of the right cylinder. Give your answers in terms of . The radius is half the diameter, or 8 ft. Top Base Bottom Base Middle What parts make-up a cylinder? 10 75

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Surface Area general • You can find the SA of any prism by using the basic formula for SA which is • L + 2 B= SA • L= lateral Surface area • L= perimeter of the base x height of the prism • B = Area of the base SA=(Lateral Area) +2(Base Area) SA= (Base Perimeter)*height+2(Area of Base) 77

The steps to SA of Prisms 1. 2. 3. 4. 5. 6. What is the base? Find perimeter of the BASE Multiply the perimeter by the height Find the area of the BASE Multiply area of BASE by 2 Add steps 3 and 5 together. 78

General SA Formula Example SA= (Base Perimeter)*height+2(Area of Base) L = Ph P = 2(9) + 2(7) = 32 ft = 32(14) = 448 ft 2 S = Ph + 2 B = 448 + 2(7)(9) = 574 ft 2 Tip! The surface area of a right rectangular prism with length ℓ, width w, and height h can be written as S = 2ℓw + 2 wh + 2ℓh. 79

Caution! The surface area formula is only true for right prisms. To find the surface area of an oblique prism, add the areas of the faces. 80

Your Turn! Part 1 Find the surface area. 81

Your Turn! Part 2 82

Surface Area of Pyramids, Cones & Spheres 83

The base of a regular pyramid is a regular polygon, and the faces are congruent isosceles triangles. The diagram shows a square pyramid. The blue dashed line labeled l is the slant height of the pyramid, the distance from the vertex to the midpoint of an edge of the base. 84

The diagram shows a cone and its net. The blue dashed line is the slant height of the cone, the distance from the vertex to a point on the edge of the base. 85

E. Right Pyramid– Surface Area Find the surface area of the pyramid. S=B+ 1 Pl 2 1 S = lw + Pl 2 S = (9 • 9) + S = 81 + 180 Use the formula. B = lw 1 (36)(10) 2 Substitute. = 4(9) = 36 P Add. S = 261 m 2 The surface area is 261 square meters. 86

F. Right Cone– Surface Area Find the surface area of the cone. Use 3. 14 for . S = r 2 + rl Use the formula. S ≈ (3. 14)(32) + (3. 14)(3)(10) Substitute. S ≈ 28. 26 + 94. 2 Multiply. S ≈ 122. 46 Add. The surface area is about 122. 46 square centimeters. 87

Your Turn! Find the surface area of each pyramid. 1 S = B + 1 Pℓ =(7)(7) + (28)(12) 2 2 S = 49 + 168 = 217 mm 2 Find the surface area of the cone. Use 3. 14 for . S = r 2 + rℓ ≈ 3. 14(2)2 + 3. 14 (2)(8) S ≈ 12. 56 + 50. 24 S ≈ 62. 8 ft 2 88

Surface area of Composite Figures 89

Warm Up Find the perimeter or circumference of each figure. Use 3. 14 for . 1. 2. 30 cm 138. 16 ft 90

Example 1: Finding Surface Areas of Composite Three-Dimensional Figures Find the surface area of the composite figure. The surface area of the rectangular prism is A right triangular prism is added to the rectangular prism. The surface area of the triangular prism is Two copies of the rectangular prism base are removed. The area of the base is B = 2(4) = 8 cm 2. 91

Example 1 Continued The surface area of the composite figure is the sum of the areas of all surfaces on the exterior of the figure. S = (rectangular prism surface area) + (triangular prism surface area) – 2(rectangular prism base area) S = 52 + 36 – 2(8) = 72 cm 2 92

Your Turn! Example 1 Find the surface area of the composite figure. Round to the nearest tenth. 93

Example 2: Recreation Application A sporting goods company sells tents in two styles, shown below. The sides and floor of each tent are made of nylon. Which tent requires less nylon to manufacture? Pup tent: Tunnel tent: The tunnel tent requires less nylon. 94

Your Turn! Example 2 A piece of ice shaped like a 5 cm by 1 cm rectangular prism has approximately the same volume as the pieces below. Compare the surface areas. Which will melt faster? Answer: The 5 cm by 1 cm prism has a surface area of 70 cm 2, which is greater than the 2 cm by 3 cm by 4 cm prism and about the same as the half cylinder. It will melt at about the same rate as the half cylinder. 95

Surface Area of Spheres 96

Warm Up Find each measurement. 1. the radius of circle M if the diameter is 25 cm 2. the circumference of circle X if the radius is 42. 5 in. 3. the area of circle T if the diameter is 26 ft 4. the circumference of circle N if the area is 625 cm 2 12. 5 cm 85 in. 169 ft 2 50 cm 97

Surface Area of a Sphere What is the volume of a sphere? ? 98

G. Sphere– Surface Area Find the surface area of a sphere with a great circle that has an area of 49 mi 2. A = r 2 49 = r 2 r = 7 Area of a circle Substitute 49 for A. Solve for r. S = 4 r 2 = 4 (7)2 = 196 mi 2 Substitute 7 for r. 99

Your Turn! Example G Find the surface area of the sphere. S = 4 r 2 Surface area of a sphere S = 4 (25)2 Substitute 25 for r. S = 2500 cm 2 100

Objectives Explore the effects of volume and surface area by changing the scale factor Vocabulary: Scale factor Assignments: 101

Finding the Scale Fcator • We can apply a , ultiplicative factor to any formula (perimeter, circumference, volume, and surface area) • Way 1: • Way 2: Plug in the multiplicator factor directly to the variable from the original formula and compare the final product to the original formula.

Example 1 A box measures 5 in. by 3 in. by 7 in. Explain whether tripling only the length, width, or height of the box would triple the volume of the box. The original box has a volume of (5)(3)(7) = 105 cm 3. What is triple the volume? 315 cm 3 A. Tripling the length: V = (15)(3)(7) = 315 cm 3 Tripling the length would triple the volume. This is the factor 103

Example Continued B. Tripling the height: V = (5)(3)(21) = 315 cm 3 Tripling the height would triple the volume. C. Tripling the width: V = (5)(9)(7) = 315 cm 3 Tripling the width would triple the volume. 104

Your Turn A cylinder measures 3 cm tall with a radius of 2 cm. Explain whether tripling only the radius or height of the cylinder would triple the amount of volume. The original cylinder has a volume of 4 • 3 = 12 cm 3. A. Tripling the radius: V = 36 • 3 = 108 cm 3 By tripling the radius, you would increase the volume nine times. B. Tripling the height: V = 4 • 9 = 36 cm 3 Tripling the height would triple the volume. 105

Example 3: Exploring Effects of Changing Dimensions original dimensions: 106

Your Turn The length, width, and height of the prism are doubled. Describe the effect on the volume. original dimensions: V = ℓwh dimensions multiplied by 2: V = ℓwh = (1. 5)(4)(3) = (3)(8)(6) = 18 = 144 Doubling the dimensions increases the volume by 8 times. 107

Example 2 What if…? Estimate the volume in gallons and the weight of the water in the aquarium if the height were doubled. Step 1 Find the volume of the aquarium in cubic feet. V = ℓwh = (120)(60)(16) = 115, 200 ft 3 Step 2 Use the conversion factor to estimate the volume in gallons. Step 3 Use the conversion factor to estimate the weight of the water. The swimming pool holds about 859, 701 gallons. The water in the swimming pool weighs about 7, 161, 313 pounds. 108

Example 2: Sports Application A sporting goods store sells exercise balls in two sizes, standard (22 -in. diameter) and jumbo (34 -in. diameter). How many times as great is the volume of a jumbo ball as the volume of a standard ball? jumbo ball: standard ball: =3. 7 A jumbo ball is about 3. 7 times as great in volume as a standard ball. 109

Your Turn! Example 2 A hummingbird eyeball has a diameter of approximately 0. 6 cm. How many times as great is the volume of a human eyeball as the volume of a hummingbird eyeball? hummingbird: human: The human eyeball is about 72. 3 times as great in volume as a hummingbird eyeball. 110

Objectives Apply concepts of density based on area and volume in modeling situations. Vocabulary: Density, population density, BTU’s Assignments: 115

Warm up – matching formulas 116

• What is Density? 117

• POPULATION DENISTY

MEASURES OF ENERGY •

Density Warm Up 1. If a metal has a mass of 154 g and has a volume of 14 cm 3, what is its density? 1. (A: 11 g/cm 3) 2. If butter has a mass of 220, what is the density of 250 m. L of butter? 2. (A: 0. 86 g/ml g) 3. (A: 43. 3 m. L) 3. If water has a density of 0. 997 g/m. L what volume must 43. 2 g of water have? 4. HONORS: A block of oak (wood) has a density of 0. 774 g/cm 3, mass of 113 g, length of 84. 5 mm, and width of 71. 0 mm. What is the thickness of this block of wood? (Hint: you may want to take 2 steps) (A: 2. 43 cm or 24. 3 mm) 120

Additional Example 1: Recreational Application A swimming pool is a rectangular prism. Estimate the volume of water in the pool in gallons when it is completely full (Hint: 1 gallon ≈ 0. 134 ft 3). The density of water is about 8. 33 pounds per gallon. Estimate the weight of the water in pounds. 121

Example 3 Continued A swimming pool is a rectangular prism. Estimate the volume of water in the pool in gallons when it is completely full (Hint: 1 gallon ≈ 0. 134 ft 3). The density of water is about 8. 33 pounds per gallon. Estimate the weight of the water in pounds. Step 1 Find the volume of the swimming pool in cubic feet. V = ℓwh = (25)(19) = 3375 ft 3 209, 804 pounds The swimming pool holds about 25, 187 gallons. The water in the swimming pool weighs about 209, 804 pounds. 122

Additional Example 2: Recreational Application 123

Additional Example 3: Recreational Application 124

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Example 2 B: Finding Lateral Areas and Surface Areas of Right Cylinders Find the lateral area and surface area of a right cylinder with circumference 24 cm and a height equal to half the radius. Give your answers in terms of . Step 1 Use the circumference to find the radius. C = 2 r 24 = 2 r r = 12 Circumference of a circle Substitute 24 for C. Divide both sides by 2. Step 2 Use the radius to find the lateral area and surface area. The height is half the radius, or 6 cm. L = 2 rh = 2 (12)(6) = 144 cm 2 Lateral area S = L + 2 r 2 = 144 + 2 (12)2 Surface area = 432 in 2 130

Your Turn! Example 2 Find the lateral area and surface area of a cylinder with a base area of 49 and a height that is 2 times the radius. Step 1 Use the circumference to find the radius. A = r 2 49 = r 2 r=7 Area of a circle Substitute 49 for A. Divide both sides by and take the square root. Step 2 Use the radius to find the lateral area and surface area. The height is twice the radius, or 14 cm. L = 2 rh = 2 (7)(14)=196 in 2 Lateral area S = L + 2 r 2 = 196 + 2 (7)2 =294 in 2 Surface area 131

Classwork/Homework Practice more volume 2, 3, 4, 5, 7, 10, 12, 16 132

Objectives Apply Geometry methods to solve design problems Vocabulary: Altitude. surface area Assignments: 133

What is a constraint? Constraints give you a value to be substituted into an equation relating a dimension to a volume or surface area, allowing the equation to be solved, and the dimensions found.

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Determining dimensions given a volume

Maximizing Volume