121616949-math.237 - 9.10 Figure 9.21 Surface Area 223...

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9.10 Surface Area 223 Figure 9.21 Approximating a surface (left) by portions of cones (right). composed of many “truncated cones;” a truncated cone is called a frustum of a cone. Figure 9.21 illustrates this approximation. So we need to be able to compute the area of a frustum of a cone. Since the frustum can be formed by removing a small cone from the top of a larger one, we can compute the desired area if we know the surface area of a cone. Suppose a right circular cone has base radius r and slant height h . If we cut the cone from the vertex to the base circle and flatten it out, we obtain a sector of a circle with radius h and arc length 2 πr , as in
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Unformatted text preview: figure 9.22 . The angle at the center, in radians, is then 2 πr/h , and the area of the cone is equal to the area of the sector of the circle. Let A be the area of the sector; since the area of the entire circle is πh 2 , we have A πh 2 = 2 πr/h 2 π A = πrh. Now suppose we have a frustum of a cone with slant height h and radii r and r 1 , as in figure 9.23 . The area of the entire cone is πr 1 ( h + h ), and the area of the small cone is πr h ; thus, the area of the frustum is πr 1 ( h + h )-πr h = π (( r 1-r ) h + r 1 h ). By similar triangles, h r = h + h r 1 ....
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