Question

A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:

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

Answer 1

The correct option is C

$\frac{2}{1}$

Step 1: Given data

As per the data given in the question,

We have,

The volume of cone $=\frac{1}{2}\times$ Volume of sphere

Step 2: Formula for finding the volume of cone and sphere

Cone:
Radius of base $=r$

height $=h$

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

Sphere :

Radius $=r$

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

Step 3: Finding the value of the ratio of altitude of the cone to the radius of its base

As, the volume of cone $=\frac{1}{2}\times$ Volume of sphere.

So,

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

$\therefore$ Ratio of altitude of cone to radius of its base will be equal to:

$$\begin{gathered} =\frac{2r}{r} \\ =\frac{2}{1} \\ =2:1 \end{gathered}$$

Hence, Ratio of altitude of cone to radius of its base will be equal to $2:1$