A cone and a hemisphere have equal bases and equal volumes.
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) $\sqrt{2}: 1$

Solution:

(b) 2 : 1

Let the radii of the cone and the hemisphere be r and their respective heights be and H.

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

$\Rightarrow \frac{h}{H}=\frac{2}{1}$