Question

If a cone and a sphere have equal radii and equal volumes, what is the ratio of the diameter of the sphere to the height of the cone?

• A. $\frac{1}{1}$
• B. $\frac{1}{2}$
• C. $\frac{2}{1}$
• D. $\frac{3}{2}$

Answer 1

The correct option is B

$\frac{1}{2}$

Step 1: Given

Cone and sphere have equal radii

The volume of the cone is equal to the volume of the sphere

Let the radius of the cone and sphere be $r$ and the diameter be $d.$

Volume of cone $=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$

Volume of sphere $=\frac{4}{3}{\mathrm{\pi r}}^3$

Step 2: Find the ratio

$$\begin{gathered} \Rightarrow \frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}=\frac{4}{3}{\mathrm{\pi r}}^3 \\ \Rightarrow \mathrm{h}=4\mathrm{r} \\ \Rightarrow \mathrm{h}=2\mathrm{d} \\ \Rightarrow \frac{\mathrm{d}}{\mathrm{h}}=\frac{1}{2} \end{gathered}$$

Hence, the correct option is $\left(\mathrm{B}\right)$