A solid sphere and a solid hemisphere have the same total surface area. The ratio of their diameters is :

1. $\sqrt{2} : 3$

2. 3 : 2

3. $\sqrt{3} : 2$

4. 3 : $\sqrt{2}$

Let radius of sphere be r and radius of solid hemisphere be R.

Given,

Total surface area of solid sphere = Total surface area of solid hemisphere

∴ 4πr² = 3πR²

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

Let diameter of sphere be d = 2r and hemisphere be D = 2R.

$\Rightarrow \dfrac{2r}{2R} = \dfrac{2\sqrt{3}}{2 \times 2} \\[1em] \Rightarrow \dfrac{d}{D} = \dfrac{\sqrt{3}}{2} \\[1em] \Rightarrow d : D = \sqrt{3} : 2.$

Hence, Option 3 is the correct option.