Question:
A sphere, a cylinder, and a cone have the same diameter. The height of the cylinder and also the cone are equal to the diameter of the sphere. Find the ratio of their volumes.
Solution:
Let r be the common radius
Height of the cone = height of the cylinder = 2r
Let
$y_{1}=$ Volume of sphere $=4 / 3 \pi r^{3}$
$\mathrm{v}_{1}=$ Volume of cylinder $=\pi r^{2} \mathrm{~h}=\pi r^{2} \times 2 r$
$v_{1}=$ Volume of cone $=1 / 3 \pi r^{2} h=1 / 3 \pi r^{3}$
Now
$v_{1}: v_{2}: v_{3}=4 / 3 \pi r^{3}: 2 \pi r^{3}: 2 / 3 \pi r^{3}$
= 4:6:2 = 2:3:1