A solid sphere and a solid hemi-sphere have the same total surface area. Find the ratio between their volumes.

Let radius of sphere be R cm and hemi-sphere be r cm.

Given,

Solid sphere and a solid hemi-sphere have the same total surface area.

$\therefore 4πR^2 = 3πr^2 \\[1em] \Rightarrow \dfrac{R^2}{r^2} = \dfrac{3π}{4π} \\[1em] \Rightarrow \dfrac{R^2}{r^2} = \dfrac{3}{4} \\[1em] \Rightarrow \dfrac{R}{r} = \sqrt{\dfrac{3}{4}} \\[1em] \Rightarrow \dfrac{R}{r} = \dfrac{\sqrt{3}}{2}.$

Calculating ratio between volumes,

$\dfrac{\text{Vol. of sphere}}{\text{Vol. of hemi-sphere}} = \dfrac{\dfrac{4}{3}πR^3}{\dfrac{2}{3}πr^3} \\[1em] = \dfrac{4πR^3 \times 3}{2πr^3 \times 3} \\[1em] = 2 \times \dfrac{R^3}{r^3} \\[1em] = 2 \times \Big(\dfrac{\sqrt{3}}{2}\Big)^3 \\[1em] = 2 \times \dfrac{3\sqrt{3}}{8} \\[1em] = \dfrac{3\sqrt{3}}{4}.$

Hence, ratio between volumes = $3\sqrt{3} : 4$.