494 CHAPTER 8 Geometry 24 in. 8 in. 16 in. 8 in. 8 in. Figure 8.41 Solution a) First we will need to calculate the area of the hexagonal base of the fish tank. Notice from Fig. 8.41 that by drawing a diagonal as indicated, the base can be divided into two identical trapezoids. To determine the area of the hexagonal base, we will calculate the area of one of these trapezoids and then multiply by 2. h b b Area of one trapezoid ( ) (8)(16 8) 96 in. Area of the hexagonal base 2(96) 192 in. 1 2 1 2 1 2 2 2 = + = + = = = Now to determine the volume of the fish tank, we will use the formula for the volume of a prism, V Bh. = We already determined that the area of the base, B, is 192 in. .2 V B h 192 24 4608 in.3 = ⋅ = ⋅ = In the above calculation, the area of the base, B, was measured in square inches and the height was measured in inches. The product of square inches and inches is cubic inches, or in. .3 b) To determine the volume of the fish tank in gallons, since 1 gal 231 in. ,3 = we will divide the volume of the fish tank in cubic inches by 231. V 4608 231 19.95 gal = ≈ Thus, the volume of the fish tank is approximately 20 gal. 7 Now try Exercise 53 Example 7 Volumes Involving Prisms Determine the volume of the remaining solid after the cylinder, triangular prism, and square prism have been cut from the solid (Fig. 8.42). 2 in. 20 in. 6 in. 4 in. 4 in. 4 in. 3 in. 8 in. Figure 8.42 Solution To determine the volume of the remaining solid, first determine the volume of the rectangular solid. Then subtract the volume of the two prisms and the cylinder that were cut out. l w h r h Volume of rectangular solid 20 3 8 480 in. Volume of circular cylinder (2 )(3) (4)(3) 12 37.70 in. Volume of triangular prism area of the base height (6)(4)(3) 36 in. 3 2 2 3 1 2 3 π π π π = ⋅ ⋅ = ⋅ ⋅ = = = = = ≈ = ⋅ = =
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