Question 24·Hard·Area and Volume

The cones are similar, meaning every pair of matching lengths changes by the same factor. Since the larger height is $4$ times the smaller height, the radius also grows by $4$: $r_{\text{large}}=4r_{\text{small}}$.

A cone has volume $V=\frac13\pi r^2h$, where $r$ is its radius and $h$ is its height. The radius supplies two factors of $4$ and the height supplies one. The $\frac13$ and $\pi$ stay the same, so volume scales by the cube of the length factor: $V_{\text{large}}=4^2\cdot4\cdot V_{\text{small}}=4^3V_{\text{small}}$.

Substitute the given smaller volume, $27\pi$, into the volume relationship: $V_{\text{large}}=4^3(27\pi)$.

Type $27\cdot4^3$ into Desmos to find the coefficient of $\pi$. It displays $1728$. Keep the $\pi$, so the larger cone’s volume is $1{,}728\pi$ cubic centimeters. Choice D.