PSAT Math Multiple-Choice Question 691: Answer and Explanation

Question: 691

A right circular cylinder has a volume of 30x cubic feet, and a cube has a volume of 21x cubic feet. What is the sum of the volumes of a cone with the same height and radius as the cylinder and of a pyramid with the same length, width, and height of the cube?

• A. 7x cubic feet
• B. 10x cubic feet
• C. 17x cubic feet
• D. 51x cubic feet

Correct Answer: C

Explanation:

(C) Let's do this one in two parts. First, we have a right cylinder with a volume of 30x. We form a cone with the same height and radius as that cylinder. The formula for the volume of a cylinder is V = πr²h, while the formula for the volume of a cone is .

Notice that the volume of a cone is just the volume of a cylinder with the same dimensions. Thus, if the volume of the cylinder is 30x, the volume of a cone with the same dimensions is or 10x.

For the second part of this problem, there's a cube with a volume of 21x. We have a pyramid with the same length, width, and height as the cube. The formula for the volume of a cube is V = lwh = L³ because the length, width, and height are all the same. The formula for the volume of a pyramid is

In this case, the pyramid has the same length, width, and height as the cube, so the volume for the pyramid can be expressed as

Notice that in this case, the volume of the pyramid is just of the volume of the cube.

The volume of the cube is 21x, so the volume of the pyramid is

The question asked us the sum of the volume of the cone and the pyramid, which can be expressed by:

V = V_cone + V_pyramid = 10x + 7x = 17x

Choice (C) is correct.