10-6 Surface Area Find the surface area of each solid. Round to the nearest tenth if necessary. 1. 3. SOLUTION: The surface area S of a regular pyramid is , where SOLUTION: The radius is 7.5 cm and the height is 15 cm. L is the lateral area and B is the area of the base. Use the Pythagorean Theorem to find the slant height. Therefore, the surface area of the pyramid is 640 cm2. ANSWER: 640 cm2 Find the surface area. 2. SOLUTION: Surface area = 2(10 m)(11 m) + 2(10 m)(15 m) + 2(11 m)(15 m) = 850 m² ANSWER: ≈ 571.9 cm2 ANSWER: 850 m² 4. SOLUTION: The segment joining the points where the slant height and height intersect the base is the apothem. eSolutions Manual - Powered by Cognero Page 1 10-6 Surface Area 5. Use the Pythagorean Theorem to find the length the apothem of the base. SOLUTION: The base of the prism is a right triangle with the legs 8 ft and 6 ft long. Use the Pythagorean Theorem to find the length of the hypotenuse of the base. A central angle of the regular hexagon is so the angle formed in the triangle below is 30. , Find the surface area. Use a trigonometric ratio to find the length of a side of the hexagon s. ANSWER: 336 ft2 Find the surface area of the pyramid. Therefore, the surface area of the pyramid is about 332.6 m2. ANSWER: ≈ 332.6 m2 eSolutions Manual - Powered by Cognero Page 2 10-6 Surface Area 8. PATIO STONES A patio stone has a rectangular base that is 3 inches by 8 inches and a height of 4 inches. What is the surface area of the stone? 6. SOLUTION: SOLUTION: ANSWER: ≈ 2063.6 cm2 7. CARS Evan is buying new tire rims that are 14 inches in diameter and 6 inches wide. Determine the surface area of each rim. Round to the nearest tenth. SOLUTION: ANSWER: 571.8 in2 ANSWER: 9. ROOFING A pyramid shaped roof has a square base that is 30 feet wide and a slant height of 14 feet. How much roofing material is needed to cover the roof? SOLUTION: ROOFING A pyramid shaped roof has a square base that is 30 feet wide and a slant height of 14 feet. How much roofing material is needed to cover the roof? The amount of roofing material needed to cover the roof is the lateral area of the pyramid. L = (0.5)Pl = (0.5)(4)(30 ft)(14 ft) = 840 ft² ANSWER: 840 ft² eSolutions Manual - Powered by Cognero Page 3 10-6 Surface Area 10. TANKS A storage tank is shown at the right. Find the surface area of each solid. Round to the nearest tenth if necessary. Round answers to the nearest tenth. 11. SOLUTION: Find the length of the third side of the triangle. a. Find the lateral area of the cylinder. b. Find the lateral area of the cone. c. Find the total lateral area of the tank. SOLUTION: a. Find the lateral area of a cylinder with a radius of feet and a height of 11 feet. Now find the surface area. b. Find the lateral area of a cone with a radius of 7 feet and a slant height of 9 feet. ANSWER: 36 ft2 c. Find the total lateral area of the tank by adding the lateral area of the cylinder and the lateral area of the cone. ANSWER: a. 483.8 ft² b. 197.9 ft² c. 681.7 ft² eSolutions Manual - Powered by Cognero Page 4 10-6 Surface Area 13. 12. SOLUTION: The diameter of the base is 3.6 cm and the height of the cylinder is 1.1 cm. SOLUTION: The radius of the cone is and the height is 18 cm. Use the Pythagorean Theorem to find the slant height . The total surface area of the prism is the sum of the areas of the bases and the lateral surface area. ANSWER: ≈ 32.8 cm2 Find the surface areas of the cone. Therefore, the surface area of the cone is about 470.7 cm2. 14. SOLUTION: ANSWER: ≈ 470.7 cm2 We need to find the area of the triangle to determine the area of the bases. Use the Pythagorean Theorem to find the height of the triangles. eSolutions Manual - Powered by Cognero Page 5 10-6 Surface Area 15. SOLUTION: The base of the pyramid is a square with side length of 8 feet and the height is 10 feet. Use the Pythagorean Theorem to find the slant height of the pyramid. Use to calculate the surface area. Find the surface area of the pyramid. ANSWER: 151.9 m2 Therefore, the surface area of the pyramid is about 236.3 ft2. ANSWER: ≈ 236.3 ft2 16. SOLUTION: Use the right triangle formed by the slant height of 7, the height of 5, and the apothem and the Pythagorean eSolutions Manual - Powered by Cognero Page 6 10-6 Surface Area Theorem to find the length of the apothem of the base. Find the surface area of the pyramid. Therefore, the surface area of the pyramid is about 302.9 cm2. ANSWER: ≈ 302.9 cm2 The base of the pyramid is an equilateral triangle. The measure of each central angle is , so the angle formed in the triangle below is 60°. 17. A cone has a diameter of 3.4 centimeters, and a slant height of 6.5 centimeters. SOLUTION: The radius of the cone is . Find the surface area. Use a trigonometric ratio to find the length s of each side of the triangle. Therefore, the surface area of the cone is about 43.8 cm2. ANSWER: ≈ 43.8 cm2 Use the formulas for regular polygons to find the perimeter and area of the base. eSolutions Manual - Powered by Cognero Page 7 10-6 Surface Area 18. A rectangular prism has = 25 centimeters, w = 18 centimeters, and h = 12 centimeters. SOLUTION: Note that the surface area of the solid is the same no matter which face is the base. So, the area of the base B is about 93.5 mm2. ANSWER: Find the surface area of the pyramid. 1932 cm2 19. A regular hexagonal pyramid has a base edge of 6 millimeters and a slant height of 9 millimeters. SOLUTION: Therefore, the surface area of the pyramid is about The base of the pyramid is a regular hexagon. The perimeter of the hexagon is P = 6 × 6 or 36 mm. 255.5 mm2. ANSWER: ≈ 255.5 mm2 A central angle of the hexagon is angle formed in the triangle below is 30°. , so the Use a trigonometric ratio to find the apothem a and then find the area of the base. eSolutions Manual - Powered by Cognero Page 8 10-6 Surface Area 20. A triangular prism has h = 6 inches and a right triangle base with legs 9 inches and 12 inches. SOLUTION: Find the other side of the triangular base. Now you can find the urface area. A central angle of the square is , so the angle formed in the triangle below is 45°. ANSWER: 324 in2 21. A cylinder has a diameter of 8 inches and a height of 6.2 inches. Use a trigonometric ratio to find the length of each side of the square. SOLUTION: ANSWER: ≈ 256.4 in.2 Find the perimeter and area of the base. 22. A square pyramid has an altitude of 12 inches and a slant height of 18 inches. SOLUTION: The base of the pyramid is square. Use the Pythagorean Theorem to find the length the apothem of the base. eSolutions Manual - Powered by Cognero Find the surface area of the pyramid. Page 9 10-6 Surface Area 24. A cone has an altitude of 5 feet, and a slant height of feet. SOLUTION: The altitude of the cone is 5 feet and the slant height Therefore, the surface area of the pyramid is about 1686.0 in2. is or 9.5 feet. Use the Pythagorean Theorem to find the radius. ANSWER: ≈ 1686.0 in2 23. A cylinder has a radius of 3 millimeters and a height of 15 millimeters. SOLUTION: ANSWER: Find the surface area of the cone. ≈ 339.3 mm2 Therefore, the surface area of the cone is about 446.1 ft2. ANSWER: ≈ 446.1 ft2 25. Find the lateral area of the tent to the nearest tenth. SOLUTION: The tent is a combination of a cylinder and a cone. eSolutions Manual - Powered by Cognero Page 10 10-6 Surface Area The cone has a radius of 5 feet and a height of 6 feet. Use the Pythagorean Theorem to find the slant height of the cone. 26. Find the lateral area of the dog house with a 12 inch square cut out of one face for the door. SOLUTION: Start by finding the lateral area of the walls (rectangular prism) and then subtracting the area of the door: The cylinder has a radius of 5 feet and a height of 12 – 6 or 6 feet. Find the sum of the lateral areas of the cone and cylinder to find the lateral area of the tent. The lateral area of the walls minus the door is . Next, find the lateral area of the roof (square pyramid): Therefore, the lateral area of the tent is about 311.2 ft2. ANSWER: 311.2 ft2 The total lateral area of the dog house is the combination of the walls ( minus the door) and the roof. Combine the lateral areas of each to get the total lateral area: ANSWER: 6465.1 in² eSolutions Manual - Powered by Cognero Page 11 10-6 Surface Area Find the surface area of each composite solid. Round to the nearest tenth if necessary. 28. 27. SOLUTION: The solid is a combination of a rectangular prism and a cylinder. The base of the rectangular prism is 6 in by 4 in and the radius of the cylinder is 3 in. The height of the solid is 15 in. SOLUTION: The solid is a combination of a cube and a cylinder. The length of each side of the cube is 12 cm and the radius of the cylinder is 6 cm. The height of the solid is 12 cm. Rectangular prism: Rectangular prism: The surface area of five faces of the rectangular prism is 720 cm2. The surface area of five faces of the rectangular prism is 258 cm2. Half-cylinder: Half-cylinder: The surface area of the half-cylinder is 108π cm2 and the total surface area is about 1059.3 cm2. The surface area of the half-cylinder is 54π cm2. ANSWER: The total surface area is 258 + 54π = 427.6. 1059.3 cm2 ANSWER: 427.6 in2 eSolutions Manual - Powered by Cognero Page 12 10-6 Surface Area 29. MOUNTAINS A conical mountain has a radius of 1.6 kilometers and a height of 0.5 kilometer. What is the lateral area of the mountain? SOLUTION: The radius of the conical mountain is 1.6 kilometers and the height is 0.5 kilometers. Use the Pythagorean Theorem to find the slant height. 30. AQUARIUMS The Tower Aquarium in Henley Beach, Australia, is the world’s largest cylindrical aquarium. It reaches a height of over 40 meters and is 36 meters in diameter. Visitors ascend through a column of water as they ride a split-level glass lift up seven floors, through the center of the aquarium. What is the approximate lateral area of the outside of the aquarium? SOLUTION: Since the diameter of the cylinder is 36 m, the radius is . Find the lateral area L of the conical mountain. ANSWER: Therefore, the lateral area is about 8.4 km2. ANSWER: 8.4 km2 eSolutions Manual - Powered by Cognero Page 13 10-6 Surface Area 31. HISTORY Archaeologists recently discovered a 1500-year-old pyramid in Mexico City. The square pyramid measures 165 yards on each side and once stood 20 yards tall. What was the original lateral area of the pyramid? SOLUTION: The pyramid has a square base with sides having lengths of 165 yards and a height of 20 yards. Use 32. MONUMENTS The monolith mysteriously appeared overnight at Seattle, Washington’s Manguson Park. It is a hollow rectangular prism 9 feet tall, 4 feet wide, and 1 foot thick. a. Find the area in square feet of the structure’s surfaces that lie above the ground. b. Use dimensional analysis to find the area in square yards. the Pythagorean Theorem to find the slant height. SOLUTION: a. The area of surfaces that lie above the ground is the sum of the area of the upper base and the lateral surface area. b. Find the lateral area L of a regular pyramid. Therefore, the lateral area of the pyramid is about 28,013.6 yd2. ANSWER: a. 94 ft2 b. 10.4 yd2 ANSWER: 28,013.6 yd2 33. TEPEES The dimensions of two canvas tepees are shown in the table at the right. Including the floors, approximately how much more canvas is used to make Tepee B than Tepee A? SOLUTION: The tepees are in the shape of a right cone. To find the amount of canvas used for each tepee, we need eSolutions Manual - Powered by Cognero Page 14 10-6 Surface Area to find its lateral area. The radius of Tepee A is or 7 feet and the radius of Tepee B is or 10 feet Tepee B will use about 380.1 ft2 more canvas than Tepee A. ANSWER: about 380.1 ft2 34. DESIGN A mailer needs to hold a poster that is almost 38 inches long and has a maximum rolled diameter of 6 inches. a. Design a mailer that is a triangular prism. Sketch the mailer and its net. b. Suppose you want to minimize the surface area of the mailer. What would be the dimensions of the mailer and its surface area? Tepee A Tepee B SOLUTION: a. A triangular prism should consist of two triangles and three rectangles. They should be connected so that, when folded together they form a prism. Use the Pythagorean Theorem to find the slant height of each tepee and then find the surface area for each cone. b. In order to minimize the surface area of the triangular prism, the triangles should be equilateral, and the side lengths of the rectangles should coincide the the length of the base of the triangle and the length of the poster. The surface area will then be . To find how much more canvas is used to make Tepee B than Tepee A, subtract the surface areas. eSolutions Manual - Powered by Cognero Use trigonometry to find the area of the triangles. The diameter of the poster has a maximum of 6 in. Page 15 10-6 Surface Area which corresponds to a radius of 3 in. The total surface area can be calculated: ANSWER: a. Sample answer: For the base we have: For the height we have: b. side lengths of triangular bases, about 10.39 in.each; height, 38 in.; 1278 in2 Now calculate the area: The area of the rectangles will be the product of the area of the base of the triangle with the length of the poster. eSolutions Manual - Powered by Cognero 35. MULTI-STEP Hector is designing a glass greenhouse for a city park. He has a 40-foot by 20foot rectangular plot available. He wants the roof to be a triangular prism in which the center of the roof is 4 feet higher than the edges. The glass costs $25 per square foot, and Hector cannot spend more than $60,000 on glass. a. What is the maximum height that Hector should make the edge of the roof? b. Describe your solution process. c. What assumptions did you make? SOLUTION: a-b. Start by sketching a figure with the given information, with g representing the height of the roof. Page 16 10-6 Surface Area after the top portion has been cut by a plane parallel to the base. The ferret tent shown is a frustum of a regular pyramid. a. Describe the faces of the solid. b. Find the surface area of the frustum formed by the tent. c. Another pet tent is made by cutting the top half off of a pyramid with a height of 12 centimeters, slant height of 20 centimeters and square base with side lengths of 32 centimeters. Find the surface area of the frustum. Find the sum of the surface areas of each individual section. The rectangular section of the front and back is 2 × 20 × g or 40g ft². The sides cover 2 × 40 × g or 80g ft². The triangular tops of the front and back of the greenhouse cover 2(0.5)(4)(20) or 80 ft². The slant of the roof is . Thus, the roof covers 2(40)(10.77) or 861 ft². The total surface area is 861 + 80 + 120g ft². Hector can use up to 60,000 ÷ 25 or 2400 ft². Therefore, g is approximately 12.1. Rounding down, we get a height of 12 ft. c. Hector used the entire available plot. There was no glass used for the base. The entrance was made of glass. The top of the roof ran along the 40 ft length of the greenhouse. ANSWER: a. about 12 ft b. First, find the sum of the surface areas of each individual section. The rectangular section of the front and back is 2 × 20 × g or 40g ft². The sides cover 2 × 40 × g or 80g ft². The triangular tops of the front and back of the greenhouse cover 2(0.5)(4)(20) or 80 ft². The slant of the roof is . Thus, the roof covers 2(40)(10.77) or 861 ft². The total surface area is 861 + 80 + 120g ft². Hector can use up to 60,000 ÷ 25 or 2400 ft². Therefore, g is approximately 12.1. Rounding down, we get a height of 12 ft. c. Hector used the entire available plot. There was no glass used for the base. The entrance was made of glass. The top of the roof ran along the 40 ft length of the greenhouse. SOLUTION: a. The two bases are squares and the 4 lateral faces are trapezoids. b. Each lateral face is a trapezoid with the bases 6 in. and 17 in. and height 15 in. The area A of a trapezoid with bases b 1, b 2 and the height h is given by the formula The lateral area of the solid is The bases are squares of sides 6 in. and 17 in. respectively. Therefore, the surface area is c. When the top half of a pyramid with a height of 12 cm, slant height of 20 cm and square base with side lengths of 32 cm is cut, the height of the frustum will be 6 cm, the slant height 10 cm and the length of each side of the upper base will be 16 cm. 36. PETS A frustum is the part of a solid that remains eSolutions Manual - Powered by Cognero Page 17 10-6 Surface Area So, the surface area of five faces of the rectangular prism is 1862 cm2. The total surface area of the frustum will be Use the Pythagorean Theorem to find the length of the hypotenuse of the base of the triangular prism. ANSWER: a. 4 trapezoids, 2 squares b. 1015 in2 Triangular prism: c. 2240 cm2 37. The three-dimensional box needs to have a clear coating painted on all six faces. What is the approximate surface area of the box? So, the surface area of four faces of the triangular prism is about 962.8 cm2. Therefore, the total surface area is about 1862 + 962.8 or 2824.8 cm2. SOLUTION: This composite solid can be divided into a rectangular prism 13 cm by 21 cm by 28 cm and a triangular prism that has a right triangle with legs of 7 cm and 21 cm as the base and a height of 28 cm. The surface area of the solid is the sum of the surface areas of each prism without the area of the 21 cm by 28 cm rectangular face at which they are joined. ANSWER: 2824.8 cm2 . Rectangular prism: eSolutions Manual - Powered by Cognero Page 18 10-6 Surface Area Find the surface area of each solid. Round to the nearest tenth. length of the base b. Use trigonometry. Slant height: 38. SOLUTION: We need to determine the slant height l. Use trigonometry. Use the exact value of l to find the lateral area. Base: This value of x is the apothem, which is only half of the length of the sides of the base. Use the exact values of area. and x to find the lateral Find the surface area. Now find the surface area. ANSWER: 510.2 mm2 ANSWER: 4524.9 ft2 39. SOLUTION: We need to determine the slant height eSolutions Manual - Powered by Cognero and the Page 19 10-6 Surface Area 40. CONSTRUCTION A road roller is a construction vehicle with smooth and heavy rollers used for compacting roads and pavement. One of these rollers has a diameter of 48 inches and is 36 inches in length. What is the area covered by the roller in two full turns? area of the square-based prism is given by L = (4s)h. The base is of the triangular prism is an equilateral triangle with an altitude equal to the side of the square, or s. To find the perimeter of the triangle, first find the length of one of its sides. SOLUTION: Total area covered = 2(Lateral Area) Use the properties of the 30-60-90 right triangle to find the length of the sides. ANSWER: 10,857.3 in.2 41. SUNCATCHERS Abby makes suncatchers of glass to sell at art shows. One style of suncatcher is a right hexagonal prism with a height of 9 centimeters and each base edge of 4 centimeters. What is the surface area of each suncatcher? (Hint: First, find the length of the apothem of the base.) SOLUTION: Apothem of base of hexagon = 4sin 60º ≈ 3.464 The side opposite the 60°-angle is times greater than the side opposite the 30°-angle. So, the side opposite the 30°-angle is . The hypotenuse is twice as long as the side opposite the 30°-angle or . The perimeter of the equilateral triangle is or triangular prism is , and the lateral area of the . The two prisms have the same height. Compare the perimeters of their bases to compare their lateral areas. ANSWER: 299.1 cm2 42. WRITING IN MATH A square-based prism and a triangular prism are the same height. The base of the triangular prism is an equilateral triangle with an altitude equal in length to the side of the square. Compare the lateral areas of the prisms. SOLUTION: The lateral area of a prism is given by L = Ph. Let h represent the height of both prisms and s represent the length of a side of the square. The perimeter of the square is P = 4s, so the lateral eSolutions Manual - Powered by Cognero The perimeter of the square-based prism is greater than that of the triangular prism, since . Therefore, the lateral area of the square-based prism is greater than that of the triangular prism. ANSWER: The lateral area of the square-based prism is greater than that of the triangular prism. The square has a perimeter of 4s and the triangle has a perimeter of and . 43. REASONING A cone and a square pyramid have the same surface area. If the areas of their bases are also equal, do they have the same slant height as well? Explain. Page 20 10-6 Surface Area SOLUTION: The surface area of the cone is given by where r is the radius of the base and is the slant height. The surface area of the square pyramid is given by where s is the length of each side of the square base and is the slant height of the pyramid. Since the base of the pyramid is a square, the perimeter is P = 4s. The area of the base of the cone and the pyramid are the same, so ANSWER: They are not equal. The slant height of the cone is or about 1.13 times greater than the slant height of the square pyramid. 44. CRITIQUE ARGUMENTS Montell and Derek are finding the surface area of a cylinder with height 5 centimeters and radius 6 centimeters. Is either of them correct? Explain. . The cone and the square pyramid have the same surface areas, so set the two expressions equal. Subtract the area of the base from each side and then solve for . SOLUTION: Therefore, Derek is correct. ANSWER: Use the equal base areas to find an equivalent expression for s. Substitute for s in the expression for . Derek; sample answer: surface area of the cylinder is , so the or . 45. REASONING Classify the following statement as sometimes, always, or never true. Justify your reasoning. The surface area of a cone of radius r and height h is less than the surface area of a cylinder of radius r and height h. SOLUTION: Consider the cylinder below. The slant height of the cone is or about 1.13 times greater than the slant height of the pyramid. Therefore, they are not equal. eSolutions Manual - Powered by Cognero The surface area of this cylinder is 2πrh + 2πr2. Page 21 10-6 Surface Area Now, consider a cone with the same base and height. Now we can compare the areas of the bases. The surface area for the cone is πrl + πr2. Compare the two formulas. The values h, r, and form a triangle, so r + h must be greater than . Therefore, 2h + r is also greater than . Thus, the statement is always true. ANSWER: Always; if the heights and radii are the same, the surface area of the cylinder will be greater since it has two circular bases and additional lateral area. 46. ARGUMENTS Determine whether the following statement is true or false. Explain your reasoning. A square pyramid and a cone both have height h units and base perimeter P units. Therefore, they have the same total surface area. SOLUTION: The surface area of a square pyramid is . The surface area of a cone is The area of the circular base is greater than the area of the square base. Now, we need to compare the lateral areas. Find . For the circle, the radius, the slant height, and the height form a right triangle. For the square, half of the side, the slant height, and the height form a right triangle. We have determined that the side of the square is , so half of the side is . This value is less than r, so we know that the radius of the square is less than the radius of the circle, so the slant height of the cone is greater than the slant height of the pyramid. Now, compare the lateral areas. . We know that the perimeter of the square base is equal to the perimeter(circumference) of the circular base. We can use this information to get the side s in terms of the radius r. eSolutions Manual - Powered by Cognero Page 22 10-6 Surface Area 47. REASONING A right prism has a height of h units and a base that is an equilateral triangle of side units. Find the general formula for the total surface area of the prism. Explain your reasoning. SOLUTION: Draw the equilateral triangle. The altitude forms two 30°-60°-90° triangles. The altitude is determined to have a length of . The slant height of the cone is greater than the slant height of the pyramid, so the lateral area of the cone is greater than the lateral area of the pyramid. The lateral area and the base of the cone are greater than the lateral area and base of the pyramid, so the statement is false. ANSWER: False; the lateral area and the base of the cone are greater than the lateral area and base of the pyramid. Find the area of the triangle. The perimeter of the triangle is area. Find the surface ANSWER: the area of an equilateral triangle of side is and the perimeter of the triangle is So, the total surface area is 48. WRITING IN MATH Describe how to find the surface area of a regular polygonal pyramid with an eSolutions Manual - Powered by Cognero Page 23 10-6 Surface Area n-gon base, height h units, and an apothem of a units. SOLUTION: Use the apothem, the height, and the Pythagorean Theorem to find the slant height of the pyramid. Then find the perimeter. Finally, use to find the surface area. The area of the base B is Divide the regular n-gon for the base into congruent isosceles triangles. Each central angle of the n-gon will have a measure of , so the measure of the angle in the right triangle created by the apothem will be ÷ 2 or . The apothem will bisect the base of the isosceles triangle so if each side of the regular polygon is s, then the side of the right triangle is . Use a trigonometric ratio to find the length of a side s. Then find the perimeter by using P = n × s. Finally, use to find the surface area where B is the area of the regular n-gon and is given by ANSWER: Use the apothem, the height, and the Pythagorean Theorem to find the slant height of the pyramid. Then use the central angle of the n-gon and the apothem to find the length of one side of the n-gon. eSolutions Manual - Powered by Cognero Page 24 10-6 Surface Area 49. Olivia makes a cylinder by bending the cardboard rectangle shown below so that the 8-centimeter sides join to form the lateral face of the cylinder. Then she cuts out two cardboard circles to form the bases and attaches these to the lateral face. Which of the following is the best estimate of the surface area of the cylinder Olivia makes? A 26 cm² B 52 cm² C 144 cm² D 196 cm² SOLUTION: Start by finding the radius of the circular bases of the cylinder that will be formed by rolling up the cardboard into a cylinder. Since the longer side of the rectangle is the base edge, this dimension also doubles as the circumference of the circle. Use this to approximate the radius of the circle: 50. A cylindrical can has a circumference of 16π inches and a height of 20 inches. What is the surface area of the can in square inches? Round to the nearest tenth. SOLUTION: Start by finding the radius of the circular bases of the cylinder. Now, find the surface area of the cylinder, with a radius of 8 in. and a height of 20 in. ANSWER: 1407.4 Now, find the surface area of the newly formed cylinder, with a radius of 2.9 cm and a height of 8 cm. Therefore, the correct choice is D. ANSWER: D eSolutions Manual - Powered by Cognero Page 25 10-6 Surface Area 51. DeMarco is wrapping presents for a party. Each present is in a box, shaped like a rectangular prism, with the dimensions shown here. DeMarco plans to wrap 8 of the boxes. 52. The top of a gazebo in a park is in the shape of a regular pentagonal pyramid. Each side of the pentagon is 10 feet long. If the slant height of the roof is about 6.9 feet, what is the lateral roof area to the nearest tenth? SOLUTION: Sketch the pyramid, labeling it with the given dimensions: Which of the following is the best estimate for the least amount of wrapping paper DeMarco will need to buy? A 1728 in² B 2016 in² C 2304 in² D 2592 in² SOLUTION: Start by finding the surface area of one box: Then, since Marcos wants to wrap 8 presents, multiply this surface area by 8: The correct choice is C. The lateral area of the gazebo's roof is 172.5 ft². ANSWER: 172.5 ANSWER: C eSolutions Manual - Powered by Cognero Page 26 10-6 Surface Area 53. A model of a cone is used to demonstrate a new filter with top. To the nearest square millimeter, what is the surface area of the cone? a. What is the surface area of the first building? A 19,600 ft2 B 28,800 ft2 C 31,500 ft2 D 48,000 ft2 b. What is the surface area of the second building? A 2705 mm2 A 16,420 ft2 B 3299 mm2 B 18,720 ft2 C 8820 mm2 C 20,925 ft2 D 9368 mm2 D 38,000 ft2 SOLUTION: Begin by using the Pythagorean Theorem to find the slant height of the cone. c. Why are the surface areas of the buildings different even though the dimensions are the same? Now, use the formula for a surface area of a cone, with a radius of 21 mm and a slant height of 29 mm. The correct choice is B SOLUTION: a. The surface area of the first building is twice the area of the base plus the lateral area. b. The surface area of the second building is the area of the base plus the lateral area. The slant height is not given, so use the Pythagorean Theorem The surface area of the cone is about 3299 mm². The correct choice is B. ANSWER: B 54. MULTI-STEP There are two separate buildings next to each other. The first is in the shape of a square prism. The dimensions of the base are 100 feet by 100 feet and the height of the building is 22 feet. The second building is in the shape of a square pyramid with the same dimensions. eSolutions Manual - Powered by Cognero The correct choice is C. c. The prism has six faces and the pyramid has only five. All of the faces of the prism are rectangles, but the pyramid has four faces that are triangles and only one that is a rectangle. Page 27 10-6 Surface Area ANSWER: a. B b. C c. The prism has six faces and the pyramid has only five. All of the faces of the prism are rectangles, but the pyramid has four faces that are triangles and only one that is a rectangle. eSolutions Manual - Powered by Cognero Page 28