SAT question a07ed090 (Area and Volume)

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?

62

Difficulty:Hard

A

$R = 8 r$ and $H = 7 h$

B

$R = 8 r$ and $H = 49 h$

C

$R = 7 r$ and $H = 8 h$

D

$R = 49 r$ and $H = 8 h$

College Board · Accepted Answer

The correct answer is C. Choice C is correct. The volume of a right circular cylinder is equal to π a^2 b , where a is the radius of a base of the cylinder and b is the height of the cylinder. It'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 π r^2 h . It'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 π 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 π ( 7 r )^2 ( 8 h ) , or π ( 49 r^2 ) ( 8 h ) , which is equivalent to π ( 392 r^2 h ) , or 392 ( π r^2 h ) . This expression is equal to 392 times the volume of the cylinder shown, π 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.

CookSAT · Answer

Note: The table tests each radius-height scale pair, and v(x,y)=x^2y shows that 7r and 8h produce the 392-fold volume.

Explanation

Enter in Desmos:

Why the other choices are wrong

SAT Question a07ed090: Area and Volume answer and explanation with Desmos solution

Explanation

Enter in Desmos:

Why the other choices are wrong