Question

A cone and a hemisphere have equal bases and equal volumes. The ratio of their heights is

• A. 1 : 2
• B. 2 : 1
• C. 4 : 1
• D. $\sqrt{2}:1$
  

Answer 1

Answer: (B)

$2:1$

Let the radius of the hemisphere and a cone be $r$, since area of their bases are equal.

The volume of hemisphere, $V_h=\frac{2}{3}\pi r^3$

and volume of cone, $V_c=\frac{1}{3}\pi r^{2}h$

$\therefore \text{Height of cone,}h=\frac{V_c}{\frac{1}{3}\pi r^2}$

$=\frac{V_h}{\frac{1}{3}\pi r^2}[\because V_c=V_h]$

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

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

$=2r$

$\therefore$ Ratio of the heights of cone and hemisphere $=\frac{\text{height of cone}}{\text{height of hemisphere}}=\frac{h}{r}=\frac{2r}{r}=\frac{2}{1}=2:1$