Geometry unit 11.4
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This document provides examples and explanations for calculating the volumes of prisms and cylinders. It begins with examples of finding the volumes of different types of prisms using the formula V=lwh. Next, it covers cylinders and uses the formula V=πr^2h to calculate volumes. It also discusses how changing the dimensions of prisms and cylinders affects their volumes. The document concludes with examples of composite figures and practice problems.
Geometry unit 11.4
- 1. UNIT 11.4 VOLUME OFUNIT 11.4 VOLUME OF PRISMS AND CYLINERSPRISMS AND CYLINERS
- 2. Warm Up Find the area of each figure. Round to the nearest tenth. 1. an equilateral triangle with edge length 20 cm 2. a regular hexagon with edge length 14 m 3. a circle with radius 6.8 in. 4. a circle with diameter 14 ft 173.2 cm2 509.2 m2 145.3 in2 153.9 ft2
- 3. Learn and apply the formula for the volume of a prism. Learn and apply the formula for the volume of a cylinder. Objectives
- 5. The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. Cavalieri’s principle says that if two three- dimensional figures have the same height and have the same cross-sectional area at every level, they have the same volume. A right prism and an oblique prism with the same base and height have the same volume.
- 7. Example 1A: Finding Volumes of Prisms Find the volume of the prism. Round to the nearest tenth, if necessary. V = ℓwh Volume of a right rectangular prism = (13)(3)(5) = 195 cm3 Substitute 13 for ℓ, 3 for w, and 5 for h.
- 8. Example 1B: Finding Volumes of Prisms Find the volume of a cube with edge length 15 in. Round to the nearest tenth, if necessary. V = s3 Volume of a cube = (15)3 = 3375 in3 Substitute 15 for s.
- 9. Example 1C: Finding Volumes of Prisms Find the volume of the right regular hexagonal prism. Round to the nearest tenth, if necessary. Step 1 Find the apothem a of the base. First draw a right triangle on one base. The measure of the angle with its vertex at the center is .
- 10. Example 1C Continued So the sides are in ratio . Step 2 Use the value of a to find the base area. Find the volume of the right regular hexagonal prism. Round to the nearest tenth, if necessary. The leg of the triangle is half the side length, or 4.5 ft. Solve for a. P = 6(9) = 54 ft
- 11. Step 3 Use the base area to find the volume. Example 1C Continued Find the volume of the right regular hexagonal prism. Round to the nearest tenth, if necessary.
- 12. Check It Out! Example 1 Find the volume of a triangular prism with a height of 9 yd whose base is a right triangle with legs 7 yd and 5 yd long. Volume of a triangular prism
- 13. Example 2: Recreation 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 ft3 ). The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds.
- 14. Step 1 Find the volume of the swimming pool in cubic feet. Step 2 Use the conversion factor to estimate the volume in gallons. Example 2 Continued V = ℓwh = (25)(15)(19) = 3375 ft3
- 15. Step 3 Use the conversion factor to estimate the weight of the water. The swimming pool holds about 25,187 gallons. The water in the swimming pool weighs about 209,804 pounds. Example 2 Continued ≈ 209,804 pounds
- 16. Check It Out! 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 ft3
- 17. Step 2 Use the conversion factor to estimate the volume in gallons. Check It Out! Example 2 Continued What if…? Estimate the volume in gallons and the weight of the water in the aquarium if the height were doubled.
- 18. What if…? Estimate the volume in gallons and the weight of the water in the aquarium if the height were doubled. Check It Out! Example 2 Continued Step 3 Use the conversion factor to estimate the weight of the water.
- 19. The swimming pool holds about 859,701 gallons. The water in the swimming pool weighs about 7,161,313 pounds. What if…? Estimate the volume in gallons and the weight of the water in the aquarium if the height were doubled. Check It Out! Example 2 Continued
- 20. Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume.
- 21. Example 3A: Finding Volumes of Cylinders Find the volume of the cylinder. Give your answers in terms of π and rounded to the nearest tenth. V = πr2 h Volume of a cylinder = π(9)2 (14) = 1134π in3 ≈ 3562.6 in3
- 22. Example 3B: Finding Volumes of Cylinders Find the volume of a cylinder with base area 121π cm2 and a height equal to twice the radius. Give your answer in terms of π and rounded to the nearest tenth. Step 1 Use the base area to find the radius. Step 2 Use the radius to find the height. The height is equal to twice the radius. πr2 = 121π Substitute 121π for the base area. r = 11 Solve for r. h = 2(r) = 2(11) = 22 cm
- 23. Example 3B Continued Step 3 Use the radius and height to find the volume. Find the volume of a cylinder with base area π and a height equal to twice the radius. Give your answers in terms of π and rounded to the nearest tenth. V = πr2 h Volume of a cylinder = π(11)2 (22) = 2662π cm3 ≈ 8362.9 cm3
- 24. Check It Out! Example 3 Find the volume of a cylinder with a diameter of 16 in. and a height of 17 in. Give your answer both in terms of π and rounded to the nearest tenth. V = πr2 h Volume of a cylinder = π(8)2 (17) = 1088π in3 ≈ 3418.1 in3 Substitute 8 for r and 17 for h.
- 25. Example 4: Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by . Describe the effect on the volume. original dimensions: radius and height multiplied by :
- 26. Example 4 Continued The radius and height of the cylinder are multiplied by . Describe the effect on the volume. Notice that . If the radius and height are multiplied by , the volume is multiplied by , or .
- 27. Check It Out! Example 4 The length, width, and height of the prism are doubled. Describe the effect on the volume. original dimensions: dimensions multiplied by 2: V = ℓwh = (1.5)(4)(3) = 18 V = ℓwh = (3)(8)(6) = 144 Doubling the dimensions increases the volume by 8 times.
- 28. Example 5: 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: The base area of the regular triangular prism is: V = ℓwh = (8)(4)(5) = 160 cm3 The volume of the regular triangular prism is: The total volume of the figure is the sum of the volumes.
- 29. Check It Out! Example 5 Find the volume of the composite figure. Round to the nearest tenth. The volume of the square prism is: Find the side length s of the base: The volume of the cylinder is: The volume of the composite is the cylinder minus the rectangular prism. Vcylinder — Vsquare prism = 45π — 90 ≈ 51.4 cm3
- 30. Lesson Quiz: Part I Find the volume of each figure. Round to the nearest tenth, if necessary. 1. a right rectangular prism with length 14 cm, width 11 cm, and height 18 cm 2. a cube with edge length 22 ft 3. a regular hexagonal prism with edge length 10 ft and height 10 ft 4. a cylinder with diameter 16 in. and height 7 in. V = 2772 cm3 V = 10,648 ft3 V ≈ 2598.1 ft3 V ≈ 1407.4 in3
- 31. Lesson Quiz: Part II 5. a cylinder with base area 196π cm2 and a height equal to the diameter 6. The edge length of the cube is tripled. Describe the effect on the volume. 7. Find the volume of the composite figure. Round to the nearest tenth. V ≈ 17,241.1 cm3 The volume is multiplied by 27. 9160.9 in3
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