17.2 Surface Area of Prisms

Transcription

1 h a b c h a b c Locker LESSON 17. Surface Area of Prisms and Cylinders Texas Math Standards The student is expected to: G.11.C Apply the formulas for the total and lateral surface area of three-dimensional figures, including prisms,... cylinders,... to solve problems using appropriate units of measure. Also G.10.B Mathematical Processes G.1.F Analyze mathematical relationships to connect and communicate mathematical ideas. Language Objective 1.B,.E.3, 3.E, 3.H.3, 4.D Explain to a partner how to find the surface area of prisms and cylinders. Name Class Date 17. Surface Area of Prisms and Cylinders Essential Question: How can you find the surface area of a prism or cylinder? G.11.C Apply the formulas for the total and lateral surface area of three-dimensional figures, including prisms,... cylinders,... to solve problems using appropriate units of measure. Also G.10.B Explore h a Developing a Surface Area Formula b c h a b c Resource Locker Surface area is the total area of all the faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces. Consider the right prism shown here and the net for the right prism. Complete the figure by labeling the dimensions of the net. ENGAGE Essential Question: How can you find the surface area of a prism or a cylinder? You find the lateral area and then add twice the area of a base. PREVIEW: LESSON PERFORMANCE TASK In the net, what type of figure is formed by the lateral faces of the prism? rectangle Write an expression for the length of the base of the rectangle. a + b + c How is the base of the rectangle related to the perimeter of the base of the prism? They are equal. The lateral area L of the prism is the area of the rectangle. Write a formula for L in terms of h, a, b, and c. L = h (a + b + c) View the Engage section online. Discuss the photograph. Ask students to identify the subject of the photo and to speculate on the significance of the surface area on determining how items are packaged. 101 Then preview the Lesson Performance Task. Module Lesson Name Class Date 17. Surface Area of Prisms and Cylinders Essential Question: How can you find the surface area of a prism or cylinder? G.11.C Apply the formulas for the total and lateral surface area of three-dimensional figures, including prisms,... cylinders,... to solve problems using appropriate units of measure. Also G.10.B Explore Developing a Surface Area Formula Resource Surface area is the total area of all the faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces. Consider the right prism shown here and the net for the right prism. Complete the figure by labeling the dimensions of the net. In the net, what type of figure is formed by the lateral faces of the prism? rectangle Write an expression for the length of the base of the rectangle. a + b + c How is the base of the rectangle related to the perimeter of the base of the prism? They are equal. The lateral area L of the prism is the area of the rectangle. Write a formula for L in terms of h, a, b, and c. L = h (a + b + c) HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. Module Lesson 101 Lesson 17.

2 F G Write the formula for L in terms of P, where P is the perimeter of the base of the prism. Let B be the area of the base of the prism. Write a formula for the surface area S of the prism in terms of B and L. Then write the formula in terms of B, P, and h. S = L + B; S = Ph + B Reflect 1. Explain why the net of the lateral surface of any right prism will always be a rectangle. Sample answer: Each lateral face of any right prism is a rectangle. The net of the lateral surface of any right prism is composed of rectangles joined end-to-end. Straight angles are formed when the rectangles are joined in this manner resulting in one long rectangular shape.. Suppose a rectangular prism has length l, width w, and height h, as shown. Explain how you can write a formula for the surface area of the prism in terms of l, w, and h. Explain 1 h B l Sample answer: There are two faces with area lw, two faces with area wh, and two faces with area lh, so the surface area can be written as S = lw + wh + lh. h w Finding the Surface Area of a Prism Lateral Area and Surface Area of Right Prisms The lateral area of a right prism with height h and base perimeter P is. The surface area of a right prism with lateral area L and base area B is S = L + B, or S = Ph + B. EXPLORE Developing a Surface Area Formula INTEGRATE TECHNOLOGY Students have the option of doing the Explore activity either in the book or online. QUESTIONING STRATEGIES In a prism, how is the lateral area formula related to the surface area formula? The surface area formula consists of the lateral area plus the area of the bases. INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Have students brainstorm how to determine what three-dimensional figure can be made from a given net and how the net can be used to find the surface area of the figure. Emphasize that prisms have parallelograms for sides, and cylinders have congruent circular bases. EXPLAIN 1 Finding the Surface Area of a Prism Module Lesson PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address TEKS G.1.F, which calls for students to analyze relationships. In this lesson, students analyze three-dimensional figures to determine how they decompose into twodimensional faces, each with its own area, and to find that the sum of the areas of the faces is equal to the surface area of the figure. Since the faces of the figures are polygons or circles, the combined areas generate the lateral area and surface area formulas students will use in this lesson. QUESTIONING STRATEGIES How can you use the formula for the area of a parallelogram to find the lateral area of a prism? Because the lateral faces of a prism are parallelograms, you can use the parallelogram formula to find the areas of the lateral faces and then add them together. Surface Area of Prisms and Cylinders 10

3 QUESTIONING STRATEGIES When can the Pythagorean Theorem be used to find the area of the bases of a triangular prism? If the bases are right triangles, then the Pythagorean Theorem can be used to find the lengths of the legs of the triangles, which are necessary to find the area of the triangles. Example 1 Each gift box is a right prism. Find the total amount of paper needed to wrap each box, not counting overlap. Step 1 Find the lateral area. Lateral area formula P = (8) + (6) = 8 cm = 8(1) Multiply. = 336 c m Step Find the surface area. Surface area formula S = L + B Substitute the lateral area. = (6)(8) Simplify. = 43 c m 8 cm 1 cm 6 cm Image Credits: C Squared Studios/Photodisc/Getty Images Step 1 Find the length c of the hypotenuse of the base. Pythagorean Theorem c = a + b Substitute. = Simplify. = Take the square root of each side. c = 6 Step Find the lateral area. Lateral area formula Substitute. = 60 ( 0 ) 4 Multiply. = 100 in 10 in. 4 in. 0 in. Module Lesson COLLABORATIVE LEARNING Small Group Activity Have students work in groups to find the surface areas of various prisms and cylinders. Have students each choose a prism or a cylinder and conjecture how to find the surface area. Then have them draw and label a model or a net and describe how to find the surface area. Ask them to verify or disprove their conjectures, and present their results to the group. 103 Lesson 17.

4 Reflect Step 3 Find the surface area. Surface area formula S = L + B Substitute. = _ 4 10 Simplify. = 1440 i n 3. A gift box is a rectangular prism with length 9.8 cm, width 10. cm, and height 9.7 cm. Explain how to estimate the amount of paper needed to wrap the box, not counting overlap. Sample answer: Round each dimension to 10 cm. Then each face has an area of approximately 10 = 100 c m, and the surface area is approximately 6 (100) = 600 c m. Your Turn Each gift box is a right prism. Find the total amount of paper needed to wrap each box, not counting overlap in. 6 in. INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns Encourage students to make an organized list of the dimensions of the lateral sides and the bases of a prism as part of their plan for finding the surface area. Then have them substitute the appropriate values into the formulas for lateral area and surface area of a prism. 18 in. 5 in. 3.6 in. 8.5 in. The lateral area is. P = (18) + (5) = 46 in. So, L = 46 (5) = 30 i n. The surface area is S = L + B. B = 18 (5) = 90 i n So, S = 30 + (90) = 410 i n. Let b be the unknown length of the leg of the base. By the Pythagorean Theorem, c = a + b, so 6 = b, 36 = b, and b = Taking the square root of each side shows that b = 4.8 in. The lateral area is. P = = 14.4 in. So, L = 14.4 (8.5) = 1.4 i n. The surface 1 area is S = L + B. B = (4.8)(3.6) = 8.64 So, S = (8.64) = i n. Module Lesson DIFFERENTIATE INSTRUCTION Multiple Representations Have students work in groups to cover boxes and cylinders with wrapping paper. Ask them to cut the wrap so that it does not overlap, and have them decompose the wraps into nets that they can use to find the surface area. Have groups discuss how the nets are related to the lateral area and the surface area formulas. Surface Area of Prisms and Cylinders 104

5 EXPLAIN Finding the Surface Area of a Cylinder QUESTIONING STRATEGIES How is the height of a right cylinder used to find its surface area? The height is used to find the lateral area. The lateral area is the circumference of the base times the height. Adding the lateral area to the area of the bases gives the surface area. Explain h Finding the Surface Area of a Cylinder Lateral Area and Surface Area of Right Cylinders The lateral area of a cylinder is the area of the curved surface that connects the two bases. The lateral area of a right cylinder with radius r and height h is L = πrh. The surface area of a right cylinder with lateral area L and base area B is S = L + B, or S = πrh + πr. r h r πr Example Each aluminum can is a right cylinder. Find the amount of paper needed for the can s label and the total amount of aluminum needed to make the can. Round to the nearest tenth. Step 1 Find the lateral area. 3 cm Lateral area formula L = πrh Substitute. L = π (3) (9) Multiply. = 54π cm Step Find the surface area. Surface area formula S = L + π r Substitute the lateral area and radius. = 54π + r (3) Simplify. = 7π cm Step 3 Use a calculator and round to the nearest tenth. 9 cm The amount of paper needed for the label is the lateral area, 54π c m. The amount of aluminum needed for the can is the surface area, 7π 6. c m. Module Lesson 105 Lesson 17.

6 B Step 1 Find the lateral area. Lateral area formula L = πrh Substitute; the radius is half the diameter. = π (.5 )( ) Multiply. = 10 π in Step Find the surface area. Surface area formula S = L + π r Substitute the lateral area and radius. = 10 π + r (.5 ) in 5 in AVOID COMMON ERRORS Common errors students make when applying the surface area formula include multiplying the height of the cylinder by the area of the base; using a diameter in the formula for cylinders instead of a radius; and forgetting to include the area of both bases. Caution students to look for these errors. Simplify. =.5 π in Step 3 Use a calculator and round to the nearest tenth. The amount of paper needed for the label is the lateral area, 10 π 31.4 i n. The amount of aluminum needed for the can is the surface area,.5 π 70.7 i n. Reflect 6. In these problems, why is it best to round only in the final step of the solution? Sample answer: This results in a more accurate answer. If you round at an intermediate step, the inaccuracies may be compounded as you perform subsequent operations. Your Turn Each aluminum can is a right cylinder. Find the amount of paper needed for the can s label and the total amount of aluminum needed to make the can. Round to the nearest tenth mm 15 cm 6 cm The lateral area is L = πrh. So, L = π (6)(15) = 180π c m. The surface area is S = L + π r. So, S = 180π + π (6) = 5π c m. The amount of paper needed for the label is the lateral area, 180π c m. The amount of aluminum needed for the can is the surface area, 5π c m. 7 mm The radius of the cylinder is half the diameter, so r = 36 mm. The lateral area is L = πrh. So, L = π (36)(80) = 5760π m m. The surface area is S = L + π r. So, S = 5760π + π (36) = 835π m m. The amount of paper needed for the label is the lateral area, 5760π 18,095.6 m m. The amount of aluminum needed for the can is the surface area, 835π 6,38.6 m m. Module Lesson Surface Area of Prisms and Cylinders 106

7 EXPLAIN 3 Finding the Surface Area of a Composite Figure QUESTIONING STRATEGIES Is the surface area of a composite figure always equal to the sum of the areas of the parts of the figure? Explain. No; you must subtract the areas of any parts of the surface that are overlapping. Explain 3 Example 3 Finding the Surface Area of a Composite Figure Find the surface area of each composite figure. Round to the nearest tenth. Step 1 Find the surface area of the right rectangular prism. Surface area formula S = Ph + B Substitute. = 80 (0) + (4) (16) Simplify. = 368 f t Step A cylinder is removed from the prism. Find the lateral area of the cylinder and the area of its bases. Lateral area formula L = πrh Substitute. = π (4) (0) Simplify. = 160π ft Base area formula B = π r Substitute. = π (4) Simplify. = 16π ft Step 3 Find the surface area of the composite figure. The surface area is the sum of the areas of all surfaces on the exterior of the figure. S = (prism surface area) + (cylinder lateral area) - (cylinder base areas) = π - (16π) = π ft 4 ft 4 ft 0 ft 16 ft Module Lesson 107 Lesson 17.

8 Step 1 Find the surface area of the right rectangular prism. Surface area formula S = Ph + B Substitute. = 6 ( 5 ) + ( 9 )( 4 ) Simplify. = 0 cm Step Find the surface area of the cylinder. Lateral area formula L = πrh Substitute. = π ( )( 3 ) Simplify. = 1 π cm Surface area formula S = L + π r Substitute. = 1 π + π ( ) Simplify. = 0 π cm 3 cm 9 cm cm 5 cm 4 cm INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns Encourage students to carefully decompose a figure as part of their plan to find its surface area. Have them make an organized list of the dimensions of the lateral sides and of the bases for each figure, along with a list of those areas that are overlapping in the composite figure. Then have them write an equation for the total surface area of the parts, including subtractions for overlapping parts, and substitute the appropriate values into the formulas. Step 3 Find the surface area of the composite figure. The surface area is the sum of the areas of all surfaces on the exterior of the figure. S = (prism surface area) + (cylinder surface area) - (area of one cylinder base) = π - π ( ) = π 39.7 cm Reflect 9. Discussion A student said the answer in Part A must be incorrect since a part of the rectangular prism is removed, yet the surface area of the composite figure is greater than the surface area of the rectangular prism. Do you agree with the student? Explain. No; removing part of the rectangular prism produces a hole through the prism and this creates additional exposed area on the interior surface of the hole. Module Lesson Surface Area of Prisms and Cylinders 108

9 ELABORATE QUESTIONING STRATEGIES How do you find the surface area of a prism? You add the perimeter of the base times the height to twice the area of the base. How do you find the surface area of a cylinder? You add the circumference of the base times the height to twice the area of the base. SUMMARIZE THE LESSON What is the same about finding the surface area of a prism and a cylinder? What is different? For both a prism and a cylinder, you find the surface area by finding the lateral area and then adding twice the area of the base; the bases of prisms and cylinders are different, so finding the lateral areas and base areas will require different processes. Your Turn Find the surface area of each composite figure. Round to the nearest tenth in Elaborate 9 in 3 in 3 in 5 in 7 in The surface area of the large prism is S large = Ph + B. So, S large = (3) (5) + (9) (7) = 86 i n. The surface area of the small prism is S small = Ph + B. So, S small = (16) (3) + (5) (3) = 78 i n. The surface area of the composite figure is the surface area of the large prism plus the surface area of the small prism minus times the area of the base of the small prism. S = (5)(3) = 344 i n 7 mm 3 mm 6 mm The surface area of the large cylinder is S large = πrh + πr. So, S large = π (7) (6) + π (7) = 18π mm. The lateral area of the small prism is L small = πrh. So, L small = π (3) (6) = 36π mm. The area of each base of the small cylinder is B = πr = π 3 = 9π mm. The surface area of the composite figure is the surface area of the large cylinder plus the lateral area of the small cylinder minus times the area of the base of the small cylinder. S = 18π + 36π - (9π) = 00π 68.3 m m 1. Can the surface area of a cylinder ever be less than the lateral area of the cylinder? Explain. No. The surface area is the lateral area plus the area of the two bases. Since the area of the two bases is greater than 0, the surface area must be greater than the lateral area. 13. Is it possible to find the surface area of a cylinder if you know the height and the circumference of the base? Explain. Yes. You can use the circumference of the base to find the radius of the base. Then you can use the height, circumference, and radius in the surface area formula. 14. Essential Question Check-In How is finding the surface area of a right prism similar to finding the surface area of a right cylinder? In both cases, you can find the surface area by finding the lateral area and then adding twice the area of a base. Module Lesson LANGUAGE SUPPORT Connect Vocabulary To help students remember the vocabulary in the lesson, including lateral area and surface area, have students make note cards of several different solid figures and their lateral and surface areas. Then have them use colored pencils to mark the dimensions of each in one color, and the formulas they will use in another color. Have them label the figures with the units and show the substitutions for the formulas. Ask them to share their note cards with other students 109 Lesson 17.

10 Evaluate: Homework and Practice EVALUATE Find the lateral area and surface area of each prism cm 3 cm Online Homework Hints and Help Extra Practice 3 ft = (4) 3 = 7 ft 5 ft S = Ph + B = 7 + (5) (7) = = 14 f t L = 7 f t S = 14 f t 7 ft = (1) = 4 cm 5 cm cm The base is a right triangle, so in the area formula, b = 3 and h = 4. S = Ph + B = 4 + ( 1_ (3) (4) ) = = 36 c m L = 4 c m S = 36 c m ASSIGNMENT GUIDE Concept and Skills Explore Developing a Surface Area Formula Example 1 Finding the Surface Area of a Prism Example Finding the Surface Area of a Cylinder Example 3 Finding the Surface Area of a Composite Figure Practice Exercise 11 Exercises 1 4 Exercises 5 6 Exercises 7 10 = (0) 10 5 cm = 00 cm S = Ph + B = 00 + (5) (5) = = 50 c m L = 00 c m S = 50 c m 5 cm 10 cm 1 m m = (7) 15 = 1080 m 15 cm The base can be divided into twelve right triangles, each triangle with a height of m and a base of 6 m. B = 1_ bh (1) = 1_ (6)(10.39)(1) = m S = L + B = (374.04) m L = 1080 m S = m Module Lesson Exercise Depth of Knowledge (D.O.K.) Mathematical Processes Recall of Information 1.C Select tools 11 Skills/Concepts 1.A Everyday life 1 0 Skills/Concepts 1.B Problem solving model 1 3 Strategic Thinking 1.A Everyday life 3 Strategic Thinking 1.D Multiple representations INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Some students may benefit from a hands-on approach for finding the surface area of solids. Have students draw simple figures like prisms and cylinders and then discuss in groups how they can find the lateral areas and the surface areas. Have them include a discussion of the properties of the faces of the figures that will help them find the lateral areas or the surface areas. INTEGRATE MATHEMATICAL PROCESSES Focus on Technology Some students may benefit from using the programming features of a graphing calculator to find the surface areas of right rectangular prisms and right cylinders. Have students enter the formulas for the surface areas of these simple solids as output from a program, with the dimensions of the solids as inputs. 3 3 Strategic Thinking 1.F Analyze relationships Surface Area of Prisms and Cylinders 1030

11 AVOID COMMON ERRORS As students find the surface area of cylinders, caution them to avoid the common errors of forgetting to include the areas of both bases, or using the diameter of the base instead of the radius in the formula. Find the lateral area and surface area of the cylinder. Leave your answer in terms of π ft 4 ft 11 in. L = πrh = π (3)(4) = 4π f t S = L + π r = 4π + π (3) = 4π + 18π = 4π f t L = 4π f t S = 4π f t 7 in. L = πrh = π (5.5)(7) = 77π i n S = L + π r = 77π + π (5.5) = 77π π = 137.5π i n L = 77π i n S = 137.5π i n Find the total surface area of the composite figure. Round to the nearest tenth ft 14 ft 6 ft 14 ft 8 ft 14 ft 14 ft 8 ft 1 ft Surface Area of Cylinder L = πrh S = L + π r = π (4)(8) = 64π + π (4) = 64π f t = 96π f t Surface Area of Prism S = L + B = (44) 1 = 58 + (14)(8) = 58 f t = 75 f t 96π - π (4) π (4) f t S f t Surface Area of Cylinder L = πrh S = L + π r = π (14) (14) = 39π + π 14 = 39π f t = 784π f t Lateral Surface Area of Prism = (40) 14 = 560 f t 784π (14 6) f t S f t Module Lesson 1031 Lesson 17.

12 Find the total surface area of the composite figure. Round to the nearest tenth ft 8 cm cm 6 cm 10 cm 9 cm 1 ft Surface Area of Prism Surface Area of Prism = (4) 9 = 16 c m = (8) 0.5 = 4 f t The base is a , (3 4 5), right S = L + B triangle, so in the area formula = 4 + () () = 1 f t b = 8 and h = 6. Total Area of Cylinder S = L + B = 16 + ( 1_ (8)(6) L = πrh ) = 64 c m = π (0.5)() = π f t Lateral Surface Area of Cylinder S = L + π r L = πrh = π + π0. 5 =.5π f t = π ()(9) = 36π c m 1 +.5π - (π0. 5 ) 18.3 f t π - (π ) 35.0 c m S 18.3 f t S 35.0 c m 11. The greater the lateral area of a florescent light bulb, the more light the bulb produces. One cylindrical light bulb is 16 inches long with a 1-inch radius. Another cylindrical bulb is 3 inches long with a 3 -inch radius. Which bulb will produce more light? 4 Lateral Area of 16 inch bulb Lateral Area of 3 inch bulb L = πrh = π (1) (16) = 3π i n 1. Find the lateral and surface area of a cube with edge length 9 inches. = (36) 9 = 34 i n S = L + B = 34 + (9) (9) L = πrh = π (0.75) (3) = 34.5π i n ft ft The 3 inch bulb will produce more light. 0.5 ft 13. Find the lateral and surface area of a cylinder with base area 64π m and a height 3 meters less than the radius. Find the Radius L = πrh A = π r = π (8)(5) 64π = π r _ = 80π m 64π _ π = π r π S = L + π r 64 = r = 80π + π (8) = = 486 i n L = 34 i n S = 486 i n 8 = r h = r - 3 h = 8-3 = 08π m L = 80π m S = 08π m Module h = 5 Lesson INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking Because a cylinder has circular bases, the circumference of the bases is the perimeter of the bases. Therefore, the lateral area of the right cylinder depends on the circumference of the base. If students think about the net for a cylinder, the net includes a rectangle and two circles. That means that the rectangle must have length equal to the circumference of the base. Surface Area of Prisms and Cylinders 103

13 14. Biology Plant cells are shaped approximately like a right rectangular prism. Each cell absorbs oxygen and nutrients through its surface. Which cell can be expected to absorb at a greater rate? (Hint: 1 μm = 1 micrometer = meter) 7 µm 35 µm 10 µm 15 µm 11 µm 15 µm Surface Area of Cell 1 Surface Area of Cell S = L + B S = L + B = (90) 7 = (35)(10) = (5) 15 = (15)(11) = 630 μ m = = 780 μ m = = 1330 μ m = 1110 μ m The cell that measures 35 μm by 7 μm by 10 μm will absorb at a greater rate. 15. Find the height of a right cylinder with surface area 160π f t and radius 5 ft. S = πrh + π r 160π = π (5) h + π (5) 160π = 10πh + 50π 110π = 10πh _ 110π _ 10π = 10πh 10π 11 = h h = 11 ft 16. Find the height of a right rectangular prism with surface area 86 m, length 10 m, and width 8 m. S = Ph + B 86 = 36h + (10) (8) 86 = 36h = 36h 3.5 = h h = 3.5 m 17. Represent Real-World Problems If one gallon of paint covers 50 square feet, how many gallons of paint will be needed to cover the shed, not including the roof? If a gallon of paint costs $5, about how much will it cost to paint the walls of the shed? 1 ft Front/Back Rectangles + Left/Right Rectangles + 1 ft Top Front/Back Triangles S = (18 1) + (1 1) + ( 1_ 18 6 ) = = 88 f t 1 gal 88 f t _ 3.3 gal 50 f t Since you can t get half a gallon, 4 total gallons will be needed. 4 $5 = ft 18 ft 4 gallons; $100 Module Lesson 1033 Lesson 17.

14 18. Match the Surface Area with the appropriate coin in the table. Coin Diameter (mm) Thickness (mm) Surface Area (m m ) Penny C Nickel A Dime B Quarter D A B C D Penny L = πrh = π (9.55)(1.55) = 9.575π mm S = L + π r = 9.575π + π m m Nickel L = πrh = π (10.605)(1.95) = π mm S = L + π r = π + π m m Dime L = πrh = π (8.955)(1.35) = π mm S = L + π r = π + π m m Quarter L = πrh = π (1.13)(1.75) = 4.455π mm S = L + π r = 4.455π + π m m INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Have students brainstorm how they would find the surface area of a prism whose dimensions have all been doubled. Does the surface area double? no If not, what is the relationship? The area is 4 times as great. Have students also consider how the surface area changes if only the height of the prism changes. Ask students to use examples to justify their reasoning. 19. Algebra The lateral area of a right rectangular prism is 144 c m. Its length is three times its width, and its height is twice its width. Find its surface area. l = 3w, h = w 144 = (w + l) h 144 = (w + 3w) w 144 = 1 6w 3 = w w = 3 cm, l = 9 cm, h = 6 cm S = L + B = (9)(3) = = 198 cm 0. A cylinder has a radius of 8 cm and a height of 3 cm. Find the height of another cylinder that has a radius of 4 cm and the same surface area as the first cylinder. L = πrh = π (8) (3) = 48π cm S = L + π r = 48π + π (8) = 48π + 18π = 176π cm S = πrh + π r 176π = π (4) h + π (4) 176π = 8πh + 3π 144π = 8πh 144π 8π = 8πh 8π 18 = h Module Lesson Surface Area of Prisms and Cylinders 1034

15 JOURNAL Have students illustrate and describe how to use formula S = L + B to find the surface area of a prism and of a right cylinder. Ask them to include all of the steps as well as the substitutions they will use in the formula. H.O.T. Focus on Higher Order Thinking 1. Analyze Relationships Ingrid is building a shelter to protect her plants from freezing. She is planning to stretch plastic sheeting over the top and the ends of the frame. Assume that the triangles in the frame on the left are equilateral. Which of the frames shown will require more plastic? Explain how finding the surface area of these figures is different from finding the lateral surface area of a figure. 10 ft 10 ft 10 ft 10 ft 10 ft Surface Area of Triangular Prism (minus bottom side) = (30) 10 = 300 cm a + b = c 5 + b = b = 100 b = 75 b = 75 S = Ph + B - Square = ( 1_ (5)( 75 )) = f t The triangular-prism-shaped frame will take more plastic; In lateral surface area, the area of the bases are not used. In this case, it is not the area of the bases that need to be removed.. Communicate Mathematical Ideas Explain how to use the net of a three-dimensional figure to find its surface area. Find the area of each part of the net, then add the areas. 3. Draw Conclusions Explain how the edge lengths of a rectangular prism can be changed so that the surface area is multiplied by 9. Triple all the edge lengths. Surface Area of Half Cylinder 1_ L = 1_ (πrh) = 1_ (π(5)(10)) = 50π f t 1_ S = 1_ L + 1_ π r = 50π + 1_ (π 5 ) = 50π + 5π = 75π 35.6 ft Module Lesson 1035 Lesson 17.

16 Lesson Performance Task A manufacturer of number cubes has the bright idea of packaging them individually in cylindrical boxes. Each number cube measures inches on a side. 1. What is the surface area of each cube?. What is the surface area of the cylindrical box? Assume the cube fits snugly in the box and that the box includes a top. Use 3.14 for π. 1. The cube has 6 faces each with an area of = 4 in. Total surface area of the cube: 6 4 i n = 4 i n. The top and bottom of the cylinder are circles, each with a diameter equal to a diagonal of one side of the cube, or inches. The radius of the top and bottom is half the diameter, or inches. Area of cylinder top = π r = 3.14 ( ) = 6.8. Total area of top and bottom: 6.8 = 1.56 i n Lateral area of cylinder: πrh = (3.14) () = 1.56 i n Total surface area of cylindrical box: ( ) i n AVOID COMMON ERRORS To find the length of a diagonal of one side of the cube, students must use the Pythagorean Theorem to find h, the hypotenuse of a right triangle with -inch sides, and then must simplify the resulting square root. Here are the steps: h = + = = 8 h = 8 = = = INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Describe how you could find the volume of an empty cylindrical number-cube container. Then find that volume. Use 3.14 for π. Subtract the volume of a number cube from the volume of a cylindrical container; about 4.56 cubic inches. V (cylinder) - V (cube) = π r h - s ( 3 ) () - () = 3.14 () () - 8 = 4.56 i n 3 Module Lesson EXTENSION ACTIVITY A packaging engineer is designing a rectangular-prism-shaped container with a surface area of 64 square inches. Find the possible dimensions for at least three containers that have surface areas of 64 square inches. Possible dimensions: 4 4 4; 8 1.6; 6.5 Find the volumes of your containers. Then propose a hypothesis about the shape of a rectangular prism with the greatest volume for a given surface area. Sample answer: The rectangular prism with the greatest volume for a given surface area is a cube. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Surface Area of Prisms and Cylinders 1036