Question

A pyramid sits inside a square prism such that its base lies on one of the bases of the prism. If the edge of the base of the square is $40$ inches and the base area of the pyramid is one-fourth of the base area of the prism, then the maximum possible volume of the pyramid is ________.

• A. $\frac{400}{3}$ times the edge of the prism
• B. $\frac{400}{3}$ times the height of the prism
• C. $400$ times the height of the prism
• D. $400$ times the base area of the prism

Answer 1

The correct option is B $\frac{400}{3}$ times the height of the prism

Edge of the base of the prism $=40$ in

Area of the base of the prism $=40\times 40=1600$ sq in

Base area of the pyramid $=\frac{1}{4}\times 1600=400$ sq in

The maximum volume of the pyramid is possible when its height is equal to the height of the prism.

Let be $h$ be that height.

Volume of the pyramid $=\frac{1}{3}\times$ Base area $\times h=\frac{1}{3}\times 400\times h=\frac{400}{3}\times h$
Thus, the maximum possible volume of the pyramid is $\frac{400}{3}$ times the height of the prism.