A cone and a hemisphere have the same base and the same height. Find the ratio of their volumes.

Let radius and height of cone and hemisphere be a cm.

Volume of cone = $\dfrac{1}{3}π(\text{radius})^2\text{height} = \dfrac{1}{3}πa^3.$

Volume of hemisphere = $\dfrac{2}{3}π(\text{radius})^3 = \dfrac{4}{3}πa^3.$

$\text{Ratio} = \dfrac{\text{Vol. of cone}}{\text{Vol. of hemisphere}} \\[1em] = \dfrac{\dfrac{1}{3}πa^3}{\dfrac{2}{3}πa^3} \\[1em] = \dfrac{1}{2} \\[1em] = 1 : 2.$

Hence, ratio between volumes = 1 : 2.