CAT 2017 Quantitative Ability

Let the each side of cube of cube be a, then
CD = $\sqrt{2}\mathit{a}$∴ CQ = $\frac{\mathit{a}}{\sqrt{2}}$
Let the radius of cone be r and height be h, then $\mathit{r}=\mathit{h}\sqrt{2}$
∴ In Δ APO and Δ CQO ( Similar triangles)
$\frac{\mathit{A}\mathit{P}}{\mathit{P}\mathit{O}}=\frac{\mathit{C}\mathit{Q}}{\mathit{O}\mathit{Q}}$
$=\frac{\mathit{r}}{\mathit{h}}=\frac{\frac{\mathit{a}}{\sqrt{2}}}{\left(\mathit{h}-\mathit{a}\right)}$
⇒$\frac{\frac{\mathit{a}}{\sqrt{2}}}{\left(\mathit{h}-\mathit{a}\right)}=\sqrt{2}$
⇒ a = 2(h - a)
⇒ $\mathit{h}=\frac{3\mathit{a}}{2}$
∴ Volume of cone$=\frac{1}{3}\mathit{\pi }\times {\left(\frac{3\mathit{a}\sqrt{2}}{2}\right)}^2\times \frac{3\mathit{a}}{2}=\frac{9}{4}{\mathit{a}}^3\mathit{\pi }$
and Volume of cube = ${\mathit{a}}^3$
∴ Required ratio $=\frac{\frac{9}{4}\mathit{\pi }{\mathit{a}}^3}{{\mathit{a}}^3}=\frac{9}{4}\mathit{\pi }=225\mathit{\pi }$