Question

Question 7
A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is 1:2:3.

Answer 1

True.
Let the radius of the hemisphere be r.

Volume of the cone, $V_1=\frac{1}{3}\pi r^{2}h$

$V_1=\frac{1}{3}\pi r^2\left(r\right)$ $[\because h=r]$

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

Now, Volume of the hemisphere, $V_2=\frac{2}{3}\pi r^3$

And volume of the cylinder, $V_3=\pi r^{2}h=\pi r^2\times r=\pi r^3$ $[\because h=r]$

$\therefore V_1:V_2:V_3=\frac{1}{3}\pi r^3:\frac{2}{3}\pi r^3:\pi r^3=1:2:3$

Hence, the ratio of their volumes is 1:2:3.