A hemisphere and a cone share the same base and have equal volumes. Given that their common radius is R, determine the height of the cone.

Correct option is A

​Given :

A hemisphere and a cone have the same base radius (R).
Volumes of hemisphere and cone are equal.

Formula Used :

Volume of hemisphere
​$$V_{\text{hemisphere}} = \frac{2}{3}\pi R^3$$​​

Volume of cone
​$$V_{\text{cone}} = \frac{1}{3}\pi R^2 h$$​​
Solution :

Height of the cone = h.

Since volumes are equal:
​$$\frac{2}{3}\pi R^3 = \frac{1}{3}\pi R^2 h$$​​

2R = h
Height of the cone = 2R