Geometry Volume of Prisms and Cylinders

GeometryVolume of Prisms and Cylinders CONFIDENTIAL

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 15-minute break for every 45 minutes of typing. About how much time will it take Max to type his term paper? 1) 2) 167 min CONFIDENTIAL

Volume of Prisms and Cylinders Thevolumeof 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

Cavalieri's principle says that if two three-dimensional figure 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 CONFIDENTIAL

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

The volume of a cube with edge length s is V = s . The volume of a right rectangular prism with length l, width w, and height h is V = lwh. 3 s h w s l s CONFIDENTIAL

Finding Volumes of Prisms Find the volume of each prism. Round to the nearest tenth, if necessary. A). 8 cm 12 cm 10 cm volume of a right rectangular prism Substitute 10 for l, 12 for w, and 8 for h is V = lwh. V = lwh = (10)(12)(8) = 980 cm 3 Next Page: CONFIDENTIAL

8 cm 12 cm B). A cube with edge length 10 cm V= s Volume of a cube = 10 = 1000 cm Substitute 10 for s. 10 cm 3 3 3 Next Page: CONFIDENTIAL

C). A right regular pentagonal Prism with base edge length 5 m and height 7 m. 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° Step:1 36° 7 m tan 36° = 2.5/a The leg of the triangle is half the side length. Or 2.5 m. a = 2.5/tan 36° Solve for a. 5 m Next Page: CONFIDENTIAL

Use the value of a to find the base area. Step:2 36° Use the base area to find the volume. Step:3 7 m 5 m CONFIDENTIAL

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. 1) 157.5 yd3 CONFIDENTIAL

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 ft ) 3 120 ft 8 ft 60 ft Next Page: CONFIDENTIAL

Find the volume of the aquarium in cubic feet. Step:1 120 ft V = lwh = (120)(60)(8) = 57,600 cm 8 ft 3 60 ft Next Page: CONFIDENTIAL

120 ft Step:2 8 ft 60 ft Next Page: CONFIDENTIAL

Step:3 120 ft 8 ft The aquarium holds about 429,851 gallons. The water in the aquarium weight about 3,580,659 pounds 60 ft CONFIDENTIAL

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 2) The volume will be doubled. 60 ft CONFIDENTIAL

Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume. CONFIDENTIAL

The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = r h. 2 h h r r CONFIDENTIAL

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 Next Page: CONFIDENTIAL

2 A cylinder with a base area of 36 in and a height equal to twice the radius. B). Step:1 Use the base area to find the radius. Use the radius to find the height. The height is equal to twice the radius. Step:2 12 cm 8 cm Use the radius and height to find the volume. Step:3 CONFIDENTIAL

Now you try!c 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. 3) 1088∏ in3 CONFIDENTIAL

Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by ½. Describe the effect on the volume. 6 m 12 m CONFIDENTIAL

Now you try! 4) The length, width, and height of the prism are doubled. Describe the effect on the volume. 1.5 ft 3 ft 4 ft 4) The volume will increased 8 times. CONFIDENTIAL

Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. 5 m 9 m 6 m 8 m CONFIDENTIAL

Now you try! 5) Find the volume of the composite figure. Round to the nearest tenth. 3 cm 5 cm 5) 90 cm3 CONFIDENTIAL

Now some problems for you to practice ! CONFIDENTIAL

Assessment 1. Find the volume of each prism. B) A) 8 m 6 cm 4 cm 9 cm 6 m 1a) 216 cm3 1b) 432√3 m3 CONFIDENTIAL

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 2) 12072 pounds 9 ft CONFIDENTIAL

3. Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth. B) A) 3 m 10 ft 5 m 12 ft 3a) 360∏ ft3 3b) 45∏ m3 CONFIDENTIAL

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 7 in. 4 ft 12 ft 4a) 6 ft3. Volume decreases by 1/64 of the original volume. 4b) 756∏ in3 . The volume increases by 27 times. CONFIDENTIAL

Find the volume of each composite figure. Round to the nearest tenth. B) A) 10 in. 5 in. 6 ft 4 ft 15 in. 14 ft 4 ft 12 ft CONFIDENTIAL

Let’s review CONFIDENTIAL