The figure shown is a right circular cylinder with a radius of r and height of h . A second

Question

The figure shown is a right circular cylinder with a radius of $r$ and height of $h$ . A second right circular cylinder (not shown) has a volume that is $392$ times as large as the volume of the cylinder shown. Which of the following could represent the radius $R$ , in terms of $r$ , and the height $H$ , in terms of $h$ , of the second cylinder?

Accepted Answer

Choice C is correct. The volume of a right circular cylinder is equal to &#960; a 2 b , where a is the radius of a base of the cylinder and b is the height of the cylinder. It&rsquo;s given that the cylinder shown has a radius of r and a height of h . It follows that the volume of the cylinder shown is equal to &#960; r 2 h . It&rsquo;s given that the second right circular cylinder has a radius of R and a height of H . It follows that the volume of the second cylinder is equal to &#960; R 2 H . Choice C gives R = 7 r and H = 8 h . Substituting 7 r for R and 8 h for H in the expression that represents the volume of the second cylinder yields &#960; 7 r 2 8 h , or &#960; 49 r 2 8 h , which is equivalent to &#960; 392 r 2 h , or 392 &#960; r 2 h . This expression is equal to 392 times the volume of the cylinder shown, &#960; r 2 h . Therefore, R = 7 r and H = 8 h could represent the radius R , in terms of r , and the height H , in terms of h , of the second cylinder. Choice A is incorrect. Substituting 8 r for R and 7 h for H in the expression that represents the volume of the second cylinder yields &#960; 8 r 2 7 h , or &#960; 64 r 2 7 h , which is equivalent to &#960; 448 r 2 h , or 448 &#960; r 2 h . This expression is equal to 448 , not 392 , times the volume of the cylinder shown. Choice B is incorrect. Substituting 8 r for R and 49 h for H in the expression that represents the volume of the second cylinder yields &#960; 8 r 2 49 h , or &#960; 64 r 2 49 h , which is equivalent to &#960; 3,136 r 2 h , or 3,136 &#960; r 2 h . This expression is equal to 3,136 , not 392 , times the volume of the cylinder shown. Choice D is incorrect. Substituting 49 r for R and 8 h for H in the expression that represents the volume of the second cylinder yields &#960; 49 r 2 8 h , or &#960; 2,401 r 2 8 h , which is equivalent to &#960; 19,208 r 2 h , or 19,208 &#960; r 2 h . This expression is equal to 19,208 , not 392 , times the volume of the cylinder shown.

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