492 CHAPTER 8 Geometry Polyhedra, Prisms, and Pyramids A polyhedron (plural, polyhedra) is a closed surface formed by the union of polygonal regions. Fig. 8.39 illustrates some polyhedra. Polyhedra Vertex Edge Face (d) (c) (b) (a) Figure 8.39 Each polygonal region is called a face of the polyhedron. The line segment formed by the intersection of two faces is called an edge. The point at which two or more edges intersect is called a vertex. In Fig. 8.39(a), there are 6 faces, 12 edges, and 8 vertices. Note that: Number of vertices number of edges number of faces 2 8 12 6 2 − + = − + = This formula, credited to Leonhard Euler, is true for any polyhedron. Solution a) The capacity of each silo can be determined using the formula for the volume of a right circular cylinder, V r h. 2 π = Since the radius is half the diameter, the radii for the three silos are 6 ft, 7 ft, and 9 ft, respectively. Now let’s determine the volumes. π π π π π π π π π π π π = = ⋅ ⋅ = ⋅ ⋅ = ≈ = = ⋅ ⋅ = ⋅ ⋅ = ≈ = = ⋅ ⋅ = ⋅ ⋅ = ≈ h h h Volume of the first silo 6 40 36 40 1440 4523.89 ft Volume of the second silo 7 50 49 50 2450 7696.90 ft Volume of the third silo 9 60 81 60 4860 15, 268.14 ft 2 2 3 2 2 3 2 2 3 r r r Therefore, the total capacity of all three silos is about 4523.89 7696.90 15,268.14 27,488.93 ft3 + + ≈ b) To determine how long it takes to empty all three silos, we will divide the total capacity by 150 ft ,3 the amount fed to Cletus’s cattle per day. 27,488.93 150 183.26 ≈ Thus, the silos will be empty in about 183 days. 7 Now try Exercise 57 Learning Catalytics Keyword: Angel-SOM-8.4 (See Preface for additional details.) Euler’s Polyhedron Formula Number of vertices number of edges number of faces 2 − + = We suggest that you verify that this formula holds for Fig. 8.39(b), (c), and (d).
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