Transcript Slide 1
Geometry Volume of Prisms and Cylinders CONFIDENTIAL 1 Warm Up 1) Marcy, Rachel, and Tina went bowling. Marcy bowled 100 less than twice Rachel’s score. Tina bowled 40 more than Rachel’s score. Rachel bowled a higher score than Marcy. What is the greatest score that Tina could have bowled? 2) Max can type 40 words per minute. He estimates that his term paper contains about 5000 words, and he takes a 15minute break for every 45 minutes of typing. About how much time will it take Max to type his term paper? CONFIDENTIAL 2 Volume of Prisms and Cylinders The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. A cube built out of 27 unit cubes has a volume of 27 cubic units. Next Page: CONFIDENTIAL 3 Cavalieri's principle says that if two three-dimensional figure have the same height and have the same crosssectional 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 CONFIDENTIAL 4 Volume of a Prism The volume of a prism with base area B and height h is V = Bh. B B h h Next Page: CONFIDENTIAL 5 The volume of a right rectangular prism with length l, width w, and height h is V = lwh. The volume of a cube with edge length s is V = s 3 . s h w l s s CONFIDENTIAL 6 Finding Volumes of Prisms Find the volume of each prism. Round to the nearest tenth, if necessary. A). 8 cm 12 cm 10 cm V = lwh 3 = (10)(12)(8) = 980 cm volume of a right rectangular prism Substitute 10 for l, 12 for w, and 8 for h is V = lwh. CONFIDENTIAL Next Page: 7 8 cm 12 cm 10 cm B). A cube with edge length 10 cm 3 V= s Volume of a cube 3 = 10 = 1000 cm3 Substitute 10 for s. Next Page: CONFIDENTIAL 8 C). A right regular pentagonal Prism with base edge length 5 m and height 7 m. 36° 7m 5m Step:1 Find the apothem a of the base . First draw a right triangle on one base as shown. The measure of the angle with its vertex at the center is 360°/10 =36° tan 36° = 2.5/a a = 2.5/tan 36° The leg of the triangle is half the side length. Or 2.5 m. Solve for a. CONFIDENTIAL Next Page: 9 36° Step:2 B= 1 2 Step:3 Use the value of a to find the base area. 2.5 tan 36 25 = 31.25 tan 36 7m 5m P = 5(5) = 25 m Use the base area to find the volume. 31.25 V = Bh = 7 301.1 m3 tan 36 CONFIDENTIAL 10 Now you try! 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. CONFIDENTIAL 11 Marine Biology Application The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds. (Hint: 1 gallon = 0.134 ft3) 120 ft 8 ft 60 ft CONFIDENTIAL Next Page: 12 Step:1 Find the volume of the aquarium in cubic feet. V = lwh = (120)(60)(8) = 57,600 cm3 120 ft 8 ft 60 ft CONFIDENTIAL Next Page: 13 120 ft 8 ft 60 ft Step:2 1 gallon Use the conversion factor to estimate the 0.134 ft3 volume in gallons. 1 gallon 1 gallon 3 57,600 ft = 429,851 gallons =1 0.134 ft3 0.134 ft3 CONFIDENTIAL Next Page: 14 120 ft 8 ft Step:3 60 ft 8.33 pounds Use the conversion factor to estimate the weight of the 1 gallon water. 8.33 pounds 8.33 pounds 429,851 gallons =1 3,580,659 pounds 1 gallon 1 gallons The aquarium holds about 429,851 gallons. The water in the aquarium weight about 3,580,659 pounds CONFIDENTIAL 15 Now you try! 2) Estimate the volume in gallons and the weight of the water in the aquarium below if the height were doubled. 120 ft 8 ft 60 ft CONFIDENTIAL 16 Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume. CONFIDENTIAL 17 The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = r2 h. h h r r CONFIDENTIAL 18 Finding Volumes of Cylinders Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth. A). 12 cm 8 cm V = r2h Volume of a cylinder = 8212 Substitute 8 for r and 12 for h. = 768 cm3 2412.7 cm3 CONFIDENTIAL Next Page: 19 A cylinder with a base area of 36 equal to twice the radius. B). Step:1 Use the base area to find the radius. r2 = 36 r=6 Step:2 Substitute 36 for the base area. Solve for r. Use the radius to find the height. The height is equal to twice the radius. 12 cm 8 cm h = 2r = 2(6) = 12cm Step:3 2 in and a height Use the radius and height to find the volume. V = r2h = (6)2(12) = 432 in3 1357.2 in3 Volume of a cylinder Substitute 6 for r and 12 for h. CONFIDENTIAL 20 Now you try! 3). Find the volume of a cylinder with a diameter of 16in. and a height of 17 in. Give your answer both in terms of and rounded to the nearest tenth. CONFIDENTIAL 21 Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by ½. Describe the effect on the volume. 6m 12 m original dimensions: radius and height multiplied by 1 . 2 V = r2h V = r2h = (6)2(12) = (3)2(6) = 432 m3 =54 m3 1 Notice that 54 = (432 ). if the radius and height are multiplied by 8 1 1 3 1 , the volume is multiplied by , or . 2 2 8 22 CONFIDENTIAL Now you try! 4) The length, width, and height of the prism are doubled. Describe the effect on the volume. 1.5 ft 4 ft CONFIDENTIAL 3 ft 23 Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. The base area of the prism is 1 B = (6)(8) =24 m2. 2 The volume of the prism is V = Bh = 24(9) = 216 m3. The cylinder's diameter equals the hypotenuse of the prism's base, 10 m. So the radius is 5 m. The volume of the cylinder is V = r2h = (5)2(5) = 125 m3. The total volume of the figure is the sum of the volumes. V = 216 + 125 608.7 m3 CONFIDENTIAL 5m 9m 6m 8m 24 Now you try! 5) Find the volume of the composite figure. Round to the nearest tenth. 3 cm 5 cm CONFIDENTIAL 25 Now some problems for you to practice ! CONFIDENTIAL 26 Assessment 1. Find the volume of each prism. B) A) 8m 6 cm 4 cm 9 cm 6m CONFIDENTIAL 27 2. The world’s largest ice cream cake, built in New York City on may 25, 2004, was approximately a 19 ft by 9 ft by 2 ft rectangular prism. Estimate the volume of the ice cream cake in gallons. If the density of the ice cream cake was 4.73 pounds per gallon, estimate the weight of the cake. (Hint: 1 gallon = 0.134 cubic feet) 19 ft 2 ft 9 ft CONFIDENTIAL 28 3. Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth. B) A) 3m 5m 10 ft 12 ft CONFIDENTIAL 29 4. Describe the effect of each change on the volume of the given figure. A) The dimensions are multiplied by ¼ . B) The dimensions are tripled. 2 in. 8 ft 4 ft 7 in. 12 ft CONFIDENTIAL 30 5. Find the volume of each composite figure. Round to the nearest tenth. B) A) 6 ft 10 in. 5 in. 4 ft 15 in. 14 ft 4 ft 12 ft CONFIDENTIAL 31 Let’s review CONFIDENTIAL 32 Volume of Prisms and Cylinders The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. A cube built out of 27 unit cubes has a volume of 27 cubic units. Next Page: CONFIDENTIAL 33 Cavalieri's principle says that if two three-dimensional figure have the same height and have the same crosssectional 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 CONFIDENTIAL 34 Volume of a Prism The volume of a prism with base area B and height h is V = Bh. B B h h Next Page: CONFIDENTIAL 35 The volume of a right rectangular prism with length l, width w, and height h is V = lwh. The volume of a cube with edge length s is V = s 3 . s h w l s s CONFIDENTIAL 36 Finding Volumes of Prisms Find the volume of each prism. Round to the nearest tenth, if necessary. A). 8 cm 12 cm 10 cm V = lwh 3 = (10)(12)(8) = 980 cm volume of a right rectangular prism Substitute 10 for l, 12 for w, and 8 for h is V = lwh. CONFIDENTIAL Next Page: 37 8 cm 12 cm 10 cm B). A cube with edge length 10 cm 3 V= s Volume of a cube 3 = 10 = 1000 cm3 Substitute 10 for s. Next Page: CONFIDENTIAL 38 C). A right regular pentagonal Prism with base edge length 5 m and height 7 m. 36° 7m 5m Step:1 Find the apothem a of the base . First draw a right triangle on one base as shown. The measure of the angle with its vertex at the center is 360°/10 =36° tan 36° = 2.5/a a = 2.5/tan 36° The leg of the triangle is half the side length. Or 2.5 m. Solve for a. CONFIDENTIAL Next Page: 39 36° Step:2 B= 1 2 Step:3 Use the value of a to find the base area. 2.5 tan 36 25 = 31.25 tan 36 7m 5m P = 5(5) = 25 m Use the base area to find the volume. 31.25 V = Bh = 7 301.1 m3 tan 36 CONFIDENTIAL 40 Marine Biology Application The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds. 3 (Hint: 1 gallon = 0.134 ft ) 120 ft 8 ft 60 ft CONFIDENTIAL Next Page: 41 Step:1 Find the volume of the aquarium in cubic feet. V = lwh = (120)(60)(8) = 57,600 cm3 120 ft 8 ft 60 ft CONFIDENTIAL Next Page: 42 120 ft 8 ft 60 ft Step:2 1 gallon Use the conversion factor to estimate the 0.134 ft3 volume in gallons. 1 gallon 1 gallon 3 57,600 ft = 429,851 gallons =1 0.134 ft3 0.134 ft3 CONFIDENTIAL Next Page: 43 120 ft Step:3 8 ft 60 ft 8.33 pounds Use the conversion factor to estimate the weight of the 1 gallon water. 8.33 pounds 8.33 pounds 429,851 gallons =1 3,580,659 pounds 1 gallon 1 gallons The aquarium holds about 429,851 gallons. The water in the aquarium weight about 3,580,659 pounds CONFIDENTIAL 44 Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume. CONFIDENTIAL 45 The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = r2 h. h h r r CONFIDENTIAL 46 Finding Volumes of Cylinders Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth. A). 12 cm 8 cm V = r2h Volume of a cylinder = 8212 Substitute 8 for r and 12 for h. = 768 cm3 2412.7 cm3 CONFIDENTIAL Next Page: 47 A cylinder with a base area of 36 equal to twice the radius. B). Step:1 Use the base area to find the radius. r2 = 36 r=6 Step:2 Substitute 36 for the base area. Solve for r. Use the radius to find the height. The height is equal to twice the radius. 12 cm 8 cm h = 2r = 2(6) = 12cm Step:3 2 in and a height Use the radius and height to find the volume. V = r2h = (6)2(12) = 432 in3 1357.2 in3 Volume of a cylinder Substitute 6 for r and 12 for h. CONFIDENTIAL 48 Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by ½. Describe the effect on the volume. 6m 12 m original dimensions: radius and height multiplied by 1 . 2 V = r2h V = r2h = (6)2(12) = (3)2(6) = 432 m3 =54 m3 1 Notice that 54 = (432 ). if the radius and height are multiplied by 8 1 1 3 1 , the volume is multiplied by , or . 2 2 8 49 CONFIDENTIAL Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. The base area of the prism is 1 B = (6)(8) =24 m2. 2 The volume of the prism is V = Bh = 24(9) = 216 m3. The cylinder's diameter equals the hypotenuse of the prism's base, 10 m. So the radius is 5 m. The volume of the cylinder is V = r2h = (5)2(5) = 125 m3. The total volume of the figure is the sum of the volumes. V = 216 + 125 608.7 m3 CONFIDENTIAL 5m 9m 6m 8m 50 You did a great job today! CONFIDENTIAL 51