Question

What is the ratio of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height?

Answer 1

Given that the diameter and the height of the cylinder, cone and sphere are the same.
The volume of cylinder, `v_1 = pir_1^2 h_1`= \[\pi \left( \frac{d}{2} \right)^2 d\]

The volume of cone, = `v_2 = 1/3pir_2^2 h_2`

\[\frac{1}{3}\pi \left( \frac{d}{2} \right)^2 d\]

And the volume of sphere,= `v_3 = 4/3pir_3^3`

\[\frac{4}{3}\pi \left( \frac{d}{2} \right)^3\]

Therefore,

The ratio of their volumes,

\[v_1 = v_2 = v_3 \]

\[ \Rightarrow \pi \left( \frac{d}{2} \right)^2 d = \frac{1}{3}\pi \left( \frac{d}{2} \right)^2 d = \frac{4}{3}\pi \left( \frac{d}{2} \right)^3 \]

\[ \Rightarrow 3: 1: 2\]

Hence, the ratio is 3 : 1 : 2