Question

The volume of cone is equal to that of sphere. If the diameter of base of cone is equal to the diameter of the sphere, what is the ratio of height of cone to the diameter of the sphere?

• A. $2:1$
• B. $1:2$
• C. $3:1$
• D. $4:1$

Answer 1

The correct option is B $1:2$

$\to$ Volume of cone

$=\frac{1}{3}DH$

$=\frac{1}{3}\pi r^{2}h$

$=\frac{1}{3}\pi (D/2{)}^{2}h$

$=\frac{\pi D^{2}h}{12}\to \left(i\right)$

Volume of sphere

$=\frac{4}{3}\pi r^3$

$=\frac{4}{3}\pi (D/2{)}^3$

$=\frac{4\pi D^3}{24}\to \left(ii\right)$

Volume of cone and

sphere is equal

$\therefore \left(i\right)=\left(ii\right)$

$\therefore \frac{\pi D^{2}h}{12}=\frac{4\pi D^3}{24}$

$\therefore \frac{h}{12}=\frac{4D}{24}$

$\therefore \frac{h}{D}=\frac{4\times 12}{24}=\frac{2}{1}$

$\therefore \frac{h}{D}=\frac{2}{1}$

Ratio of height of cone to

Diameter of sphere is $2:1$