Question

A right circular cone and a sphere have equal volumes. If the radius of the base of the cone is $2x$ and the radius of the sphere is $3x$, find the height of the cone in terms of $x$.

• A. $x$
• B. $\frac{3x}{2}$
• C. $\frac{4x}{3}$
• D. $20x$
• E. $27x$

Answer 1

Answer: (E)

$27x$

Given, volumes of right circular cone and sphere are equal.

The radius of right circular cone is $2x$ and sphere is $3x$.

Let the height of right circular cone is $h$.

Then volume of right circular cone $=$ $\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi (2x{)}^{2}h=\frac{4}{3}\pi x^{2}h$.

Then volume of sphere $=$ $\frac{4}{3}\pi r^3=\frac{4}{3}\pi (3x{)}^3=\frac{4}{3}\times 27\pi x^3$

But volumes of right circular cone and sphere are equal.

$\therefore \frac{4}{3}\pi x^{2}h=\frac{4}{3}\times 27\pi x^3$

$\therefore h=27x$