Question

A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total height of the solid is 3 times the volume of the cone, then the height of the cylinder is

(a) 2h

(b) $\frac{3h}{2}$

(c) $\frac{h}{2}$

(d) $\frac{2h}{2}$

Answer 1

Disclaimer: In the the question, the statement given is incorrect. Instead of total height of solid being equal to 3 times the volume
of cone, the volume of the total solid should be equal to 3 times the volume of the cone.

Let x be the height of cylinder.

Since, volume of the total solid should be equal to 3 times the volume of the cone,
So,
$$\begin{gathered} \frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}+{\mathrm{\pi r}}^2\mathrm{x}=3\left(\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}\right) \\ \Rightarrow \frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}-{\mathrm{\pi r}}^2\mathrm{h}+{\mathrm{\pi r}}^2\mathrm{x}=0 \\ \Rightarrow {\mathrm{\pi r}}^2\mathrm{x}=\frac{2}{3}{\mathrm{\pi r}}^2\mathrm{h} \\ \Rightarrow \mathrm{x}=\frac{2}{3}\mathrm{h} \end{gathered}$$

Hence, the height of cylindrical part

Hence, the correct answer is choice (d).