# 12-4 Volumes of Prisms and Cylinders. Find the volume of each prism.

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1 Find the volume of each prism. 3. the oblique rectangular prism shown at the right 1. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. The volume is 108 cm cm 3 The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism m 3 4. an oblique pentagonal prism with a base area of 42 square centimeters and a height of 5.2 centimeters If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same cm in 3 esolutions Manual - Powered by Cognero Page 1

2 Find the volume of each cylinder. Round to the nearest tenth. 7. a cylinder with a diameter of 16 centimeters and a height of 5.1 centimeters cm ft 3 8. a cylinder with a radius of 4.2 inches and a height of 7.4 inches 6. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right cylinder and an oblique one of the same height and cross sectional area are same in 3 9. MULTIPLE CHOICE A rectangular lap pool measures 80 feet long by 20 feet wide. If it needs to be filled to four feet deep and each cubic foot holds 7.5 gallons, how many gallons will it take to fill the lap pool? A 4000 B 6400 C 30,000 D 48, m 3 Each cubic foot holds 7.5 gallons of water. So, the amount of water required to fill the pool is 6400(7.5) = 48,000. Therefore, the correct choice is D. D esolutions Manual - Powered by Cognero Page 2

3 CCSS SENSE-MAKING Find the volume of each prism. 10. The base is a rectangle of length 3 in. and width 2 in. The height of the prism is 5 in. 12. The base is a right triangle with a leg length of 9 cm and the hypotenuse of length 15 cm. Use the Pythagorean Theorem to find the height of the base. 30 in 3 The height of the prism is 6 cm. 11. The base is a triangle with a base length of 11 m and the corresponding height of 7 m. The height of the prism is 14 m. 324 cm m 3 esolutions Manual - Powered by Cognero Page 3

4 13. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. B = 11.4 ft 2 and h = 5.1 ft. Therefore, the volume is ft an oblique hexagonal prism with a height of 15 centimeters and with a base area of 136 square centimeters If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. 15. a square prism with a base edge of 9.5 inches and a height of 17 inches 16. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same in 3 CCSS SENSE-MAKING Find the volume of each cylinder. Round to the nearest tenth. r = 5 yd and h = 18 yd yd cm 3 esolutions Manual - Powered by Cognero Page 4

5 17. r = 6 cm and h = 3.6 cm cm If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same. r = 7.5 mm and h = 15.2 mm. 18. r = 5.5 in. Use the Pythagorean Theorem to find the height of the cylinder mm PLANTER A planter is in the shape of a rectangular prism 18 inches long, inches deep, and 12 inches high. What is the volume of potting soil in the planter if the planter is filled to below the top? inches The planter is to be filled inches below the top, so Now you can find the volume in in 3 esolutions Manual - Powered by Cognero Page 5

6 21. SHIPPING A box 18 centimeters by 9 centimeters by 15 centimeters is being used to ship two cylindrical candles. Each candle has a diameter of 9 centimeters and a height of 15 centimeters, as shown at the right. What is the volume of the empty space in the box? 22. SANDCASTLES In a sandcastle competition, contestants are allowed to use only water, shovels, and 10 cubic feet of sand. To transport the correct amount of sand, they want to create cylinders that are 2 feet tall to hold enough sand for one contestant. What should the diameter of the cylinders be? V = 10 ft 3 and h = 2 ft Use the formula to find r. The volume of the empty space is the difference of volumes of the rectangular prism and the cylinders. Therefore, the diameter of the cylinders should be about 2.52 ft ft cm 3 esolutions Manual - Powered by Cognero Page 6

7 Find the volume of the solid formed by each net. 23. The middle piece of the net is the front of the solid. The top and bottom pieces are the bases and the pieces on the ends are the side faces. This is a triangular prism. One leg of the base 14 cm and the hypotenuse 31.4 cm. Use the Pythagorean Theorem to find the height of the base. The height of the prism is 20 cm. The volume V of a prism is V = Bh, where B is the area of the base, h is the height of the prism. 24. The circular bases at the top and bottom of the net indicate that this is a cylinder. If the middle piece were a rectangle, then the prism would be right. However, since the middle piece is a parallelogram, it is oblique. The radius is 1.8 m, the height is 4.8 m, and the slant height is 6 m. If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume. So, the volume of a right prism and an oblique one of the same height and cross sectional area are same cm m 3 esolutions Manual - Powered by Cognero Page 7

9 a. The volume is multiplied by 3. b. The volume is multiplied by 3 2 or 9. c. The volume is multiplied by 3 3 or 27. d. The volume is multiplied by 27. SOIL A soil scientist wants to determine the bulk density of a potting soil to assess how well a specific plant will grow in it. The density of the soil sample is the ratio of its weight to its volume. a. If the weight of the container with the soil is 20 pounds and the weight of the container alone is 5 pounds, what is the soil s bulk density? b. Assuming that all other factors are favorable, how well should a plant grow in this soil if a bulk density of pound per square inch is desirable for root growth? Explain. c. If a bag of this soil holds 2.5 cubic feet, what is its weight in pounds? a lb / in 3 b. The plant should grow well in this soil since the bulk density of lb / in 3 is close to the desired bulk density of lb / in 3. c. 8.3 lb Find the volume of each composite solid. Round to the nearest tenth if necessary. a. First calculate the volume of soil in the pot. Then divide the weight of the soil by the volume. 28. The solid is a combination of two rectangular prisms. The base of one rectangular prism is 5 cm by 3 cm and the height is 11 cm. The base of the other prism is 4 cm by 3 cm and the height is 5 cm. The weight of the soil is the weight of the pot with soil minus the weight of the pot. W = 20 5 = 15 lbs. 225 cm 3 The soil density is thus: b lb/in 3 is close to lb/in 3 so the plant should grow fairly well. c. esolutions Manual - Powered by Cognero Page 9

10 31. MANUFACTURING A can 12 centimeters tall fits into a rubberized cylindrical holder that is 11.5 centimeters tall, including 1 centimeter for the thickness of the base of the holder. The thickness of the rim of the holder is 1 centimeter. What is the volume of the rubberized material that makes up the holder? 29. The solid is a combination of a rectangular prism and a right triangular prism. The total volume of the solid is the sum of the volumes of the two rectangular prisms. The volume of the rubberized material is the difference between the volumes of the container and the space used for the can. The container has a radius of and a height of 11.5 cm. 120 m 3 The empty space used to keep the can has a radius of 3.25 cm and a height of = 10.5 cm. The volume V of a cylinder is V = πr 2 h, where r is the radius of the base and h is the height of the cylinder. 30. Therefore, the volume of the rubberized material is about cm 3. The solid is a combination of a rectangular prism and two half cylinders cm in 3 esolutions Manual - Powered by Cognero Page 10

11 Find each measure to the nearest tenth. 32. A cylindrical can has a volume of 363 cubic centimeters. The diameter of the can is 9 centimeters. What is the height? 33. A cylinder has a surface area of 144π square inches and a height of 6 inches. What is the volume? Use the surface area formula to solve for r. 5.7 cm The radius is 6. Find the volume in 3 esolutions Manual - Powered by Cognero Page 11

12 34. A rectangular prism has a surface area of 432 square inches, a height of 6 inches, and a width of 12 inches. What is the volume? Use the surface area formula to find the length of the base of the prism. Find the volume. 35. ARCHITECTURE A cylindrical stainless steel column is used to hide a ventilation system in a new building. According to the specifications, the diameter of the column can be between 30 centimeters and 95 centimeters. The height is to be 500 centimeters. What is the difference in volume between the largest and smallest possible column? Round to the nearest tenth cubic centimeter. The volume will be the highest when the diameter is 95 cm and will be the lowest when it is 30 cm.that is when the radii are 47.5 cm and 15 cm respectively. Find the difference between the volumes. 576 in 3 3,190,680.0 cm 3 esolutions Manual - Powered by Cognero Page 12

13 36. CCSS MODELING The base of a rectangular swimming pool is sloped so one end of the pool is 6 feet deep and the other end is 3 feet deep, as shown in the figure. If the width is 15 feet, find the volume of water it takes to fill the pool. 37. CHANGING DIMENSIONS A soy milk company is planning a promotion in which the volume of soy milk in each container will be increased by 25%. The company wants the base of the container to stay the same. What will be the height of the new containers? The swimming pool is a combination of a rectangular prism and a trapezoidal prism. The base of the rectangular prism is 6 ft by 10 ft and the height is 15 ft. The bases of the trapezoidal prism are 6 ft and 3 ft long and the height of the base is 10 ft. The height of the trapezoidal prism is 15 ft. The total volume of the solid is the sum of the volumes of the two prisms. Find the volume of the original container. The volume of the new container is 125% of the original container, with the same base dimensions. Use 1.25V and B to find h ft DESIGN Sketch and label (in inches) three different designs for a dry ingredient measuring cup that holds 1 cup. Be sure to include the dimensions in each drawing. (1 cup in 3 ) Sample answers: For any cylindrical container, we have the following equation for volume: esolutions Manual - Powered by Cognero Page 13

14 The last equation gives us a relation between the radius and height of the cylinder that must be fulfilled to get the desired volume. First, choose a suitable radius, say 1.85 in, and solve for the height. If we choose a height of say 4 in., then we can solve for the radius. Sample answers: For any rectangular container, the volume equation is: Choose numbers for any two of the dimensions and we can solve for the third. Let l = 2.25 in. and w = 2.5 in. esolutions Manual - Powered by Cognero Page 14

15 39. Find the volume of the regular pentagonal prism by dividing it into five equal triangular prisms. Describe the base area and height of each triangular prism. The base of the prism can be divided into 5 congruent triangles of a base 8 cm and the corresponding height 5.5 cm. So, the pentagonal prism is a combination of 5 triangular prisms of height 10 cm. Find the base area of each triangular prism. 40. PATIOS Mr. Thomas is planning to remove an old patio and install a new rectangular concrete patio 20 feet long, 12 feet wide, and 4 inches thick. One contractor bid \$2225 for the project. A second contractor bid \$500 per cubic yard for the new patio and \$700 for removal of the old patio. Which is the less expensive option? Explain. Convert all of the dimensions to yards. 20 feet = yd 12 feet = 4 yd 4 in. = yd Find the volume. Therefore, the volume of the pentagonal prism is 1100 cm 3 ; Each triangular prism has a base area of or 22 cm 2 and a height of 10 cm. The total cost for the second contractor is about. Therefore, the second contractor is a less expensive option. Because 2.96 yd 3 of concrete are needed, the second contractor is less expensive at \$ MULTIPLE REPRESENTATIONS In this problem, you will investigate right and oblique cylinders. a. GEOMETRIC Draw a right cylinder and an oblique cylinder with a height of 10 meters and a diameter of 6 meters. b. VERBAL A square prism has a height of 10 meters and a base edge of 6 meters. Is its volume greater than, less than, or equal to the volume of the cylinder? Explain. c. ANALYTICAL Describe which change affects esolutions Manual - Powered by Cognero Page 15

16 the volume of the cylinder more: multiplying the height by x or multiplying the radius by x. Explain. a. The oblique cylinder should look like the right cylinder (same height and size), except that it is pushed a little to the side, like a slinky. b. Find the volume of each. b. Greater than; a square with a side length of 6 m has an area of 36 m 2. A circle with a diameter of 6 m has an area of 9π or 28.3 m 2. Since the heights are the same, the volume of the square prism is greater. c. Multiplying the radius by x ; since the volume is represented by π r 2 h, multiplying the height by x makes the volume x times greater. Multiplying the radius by x makes the volume x 2 times greater, assuming x > CCSS CRITIQUE Franciso and Valerie each calculated the volume of an equilateral triangular prism with an apothem of 4 units and height of 5 units. Is either of them correct? Explain your reasoning. The volume of the square prism is greater. c. Do each scenario. Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct. Francisco used the standard formula for the volume of a solid, V = Bh. The area of the base, B, is onehalf the apothem multiplied by the perimeter of the base. Assuming x > 1, multiplying the radius by x makes the volume x 2 times greater. For example, if x = 0.5, then x 2 = 0.25, which is less than x. a. Francisco; Valerie incorrectly used as the length of one side of the triangular base. Francisco used a different approach, but his solution is correct. 43. CHALLENGE A cylindrical can is used to fill a esolutions Manual - Powered by Cognero Page 16

17 container with liquid. It takes three full cans to fill the container. Describe possible dimensions of the container if it is each of the following shapes. a. rectangular prism b. square prism c. triangular prism with a right triangle as the base 3 by 4 by 10π Sample answers: a. 3 by 5 by 4π The volume of the can is 20π in 3. It takes three full cans to fill the container, so the volume of the container is 60π in 3. a. Choose some basic values for 2 of the sides, and then determine the third side. Base: 3 by 5. 3 by 5 by 4π b. Choose some basic values for 2 of the sides, and then determine the third side. Base: 5 by 5. b. 5 by 5 by c. base with legs measuring 3 in. and 4 in., height 10π in. 44. WRITING IN MATH Write a helpful response to the following question posted on an Internet gardening forum. I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order? Sample answer: The nursery means a cubic yard, which is 3 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. Sample answer: The nursery means a cubic yard, which is 3 3 or 27 cubic feet. Find the volume of your garden in cubic feet and divide by 27 to determine the number of cubic yards of soil needed. 5 by 5 by c. Choose some basic values for 2 of the sides, and then determine the third side. Base: Legs: 3 by 4. esolutions Manual - Powered by Cognero Page 17

18 45. OPEN ENDED Draw and label a prism that has a volume of 50 cubic centimeters. Choose 3 values that multiply to make 50. The factors of 50 are 2, 5, 5, so these are the simplest values to choose. Sample answer: Sample answer: 46. REASONING Determine whether the following statement is true or false. Explain. Two cylinders with the same height and the same lateral area must have the same volume. True; if two cylinders have the same height (h 1 = h 2 ) and the same lateral area (L 1 = L 2 ), the circular bases must have the same area. 47. WRITING IN MATH How are the formulas for the volume of a prism and the volume of a cylinder similar? How are they different? Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr 2. Sample answer: Both formulas involve multiplying the area of the base by the height. The base of a prism is a polygon, so the expression representing the area varies, depending on the type of polygon it is. The base of a cylinder is a circle, so its area is πr The volume of a triangular prism is 1380 cubic centimeters. Its base is a right triangle with legs measuring 8 centimeters and 15 centimeters. What is the height of the prism? A 34.5 cm B 23 cm C 17 cm D 11.5 cm B The radii must also be equal. True; if two cylinders have the same height and the same lateral area, the circular bases must have the same area. Therefore, πr 2 h is the same for each cylinder. esolutions Manual - Powered by Cognero Page 18

19 49. A cylindrical tank used for oil storage has a height that is half the length of its radius. If the volume of the tank is 1,122,360 ft 3, what is the tank s radius? F 89.4 ft G ft H ft J ft 51. SAT/ACT A county proposes to enact a new 0.5% property tax. What would be the additional tax amount for a landowner whose property has a taxable value of \$85,000? A \$4.25 B \$170 C \$425 D \$4250 E \$42,500 Find the 0.5% of \$85,000. Therefore, the correct choice is C. C Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary. F 50. SHORT RESPONSE What is the ratio of the area of the circle to the area of the square? The radius of the circle is 2x and the length of each side of the square is 4x. So, the ratio of the areas can be written as shown. 52. The lateral area L of a regular pyramid is, where is the slant height and P is the perimeter of the base. The slant height is the height of each of the congruent lateral triangular faces. Use the Pythagorean Theorem to find the slant height. esolutions Manual - Powered by Cognero Page 19

20 Find the perimeter and area of the equilateral triangle for the base. Use the Pythagorean Theorem to find the height h of the triangle. Therefore, the surface area of the pyramid is about ft ft 2 ; ft 2 The perimeter is P = 3 10 or 30 feet. So, the area of the base B is ft 2. Find the lateral area L and surface area S of the regular pyramid. So, the lateral area of the pyramid is about ft 2. esolutions Manual - Powered by Cognero Page 20

21 angle formed in the triangle below is The lateral area L of a regular pyramid is, where is the slant height and P is the perimeter of the base. Here, the base is a square of side 7 cm and the slant height is 9 cm. Use a trigonometric ratio to find the measure of the apothem a. So, the lateral area of the pyramid is 126 cm 2. Find the lateral area and surface area of the pyramid. The surface area S of a regular pyramid is, where L is the lateral area and B is the area of the base. So, the lateral area of the pyramid is in 2. Therefore, the surface area of the pyramid is 175 cm cm 2 ; 175 cm 2 Therefore, the surface area of the pyramid is about in The pyramid has a slant height of 15 inches and the base is a hexagon with sides of 10.5 inches. A central angle of the hexagon is or 60, so the angle formed in the triangle below is in 2 ; in 2 esolutions Manual - Powered by Cognero Page 21

22 55. BAKING Many baking pans are given a special nonstick coating. A rectangular cake pan is 9 inches by 13 inches by 2 inches deep. What is the area of the inside of the pan that needs to be coated? 57. Find the diameter of a circle with an area of 102 square centimeters. The area that needs to be coated is the sum of the lateral area and one base area. Therefore, the area that needs to be coated is 2(13 + 9)(2) + 13(9) = 205 in in 2 Find the indicated measure. Round to the nearest tenth. 56. The area of a circle is 54 square meters. Find the diameter. The diameter of the circle is about 11.4 m cm 58. The area of a circle is 191 square feet. Find the radius. The diameter of the circle is about 8.3 m. 7.8 ft 8.3 m esolutions Manual - Powered by Cognero Page 22

23 59. Find the radius of a circle with an area of 271 square inches. 61. The area A of a trapezoid is one half the product of the height h and the sum of the lengths of its bases, b 1 and b in. Find the area of each trapezoid, rhombus, or kite. 378 m The area A of a kite is one half the product of the lengths of its diagonals, d 1 and d d 1 = 2(22) = 44 and d 2 = 2(23) = 46 d 1 = 12 in. and d 2 = = 20 in ft in 2 esolutions Manual - Powered by Cognero Page 23

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