Question

A right circular cone and a sphere have equal volumes. If the cone has radius $x$ and height $2x$, calculate the radius of the sphere.

• A. $x$
• B. $\frac{x}{{}^3\sqrt{2}}$
• C. ${}^3\sqrt{2}$
• D. $\frac{1}{{}^3\sqrt{2}}$
• E. ${}^3\sqrt{2x}$

Answer 1

The correct option is B $\frac{x}{{}^3\sqrt{2}}$

Given, volume of right circular cone and sphere is equal.

The radius $x$ and height is $2x$ of right circular cone.

Then volume of right circular cone $=$ $\frac{1}{3}\pi \times x^{2}2x=\frac{2}{3}\pi r^3$

Let $r$ is the radius of sphere

Then volume of sphere $=$$\frac{4}{3}\pi r^3$

But volume of right circular cone and sphere is equal

$\therefore \frac{4}{3}\pi r^3=\frac{2}{3}\pi x^3$

$\Rightarrow r^3=\frac{1}{2}{x}^3$

$\Rightarrow r=\frac{x}{\sqrt[3]{32}}$

Then radius of sphere $=$ $\frac{x}{\sqrt[3]{32}}$