Question

A cylinder and a cone have the same volume. The cylinder has radius x and height y. The cone has radius 2x. Find the height of the cone in terms of y.

Answer 1

$\bf \textit{volume of a cylinder}\\\\ V_c=\pi r^2h\qquad \begin{cases} r=radius\\ h=height\\ ------\\ r=x\\ h=y \end{cases}\implies V_c=x^2y\pi \\\\\\ \textit{volume of a cone}\\\\ V_n=\cfrac{\pi r^2h}{3}\qquad r=2x\implies V_n=\cfrac{\pi \cdot (2x)^2h}{3} \\\\\\ V_n=\cfrac{\pi \cdot 2^2x^2h}{3}\implies V_n=\cfrac{4x^2h\pi }{3}\\\\ -------------------------------$

$\bf V_c=V_n\implies x^2y\pi =\cfrac{4x^2h\pi }{3}\implies 3x^2y\pi =4x^2h\pi \\\\\\ \cfrac{3x^2y\pi }{4x^2\pi }=h\implies \boxed{\cfrac{3y}{4}=h}$