If the ratio of the radii of the sphere and hemisphere is $\sqrt{3}:2$, then determine the ratio of their total surface area?

1:1

The correct answer is Option (4) → 1:1

Let the radius of the sphere be $r_1$ and that of the hemisphere be $r_2$.

Given:

$\frac{r_1}{r_2} = \frac{\sqrt{3}}{2}$

Surface area of a sphere = $4\pi r_1^2$

Total surface area of a hemisphere = $3\pi r_2^2$

Required ratio = $\frac{4\pi r_1^2}{3\pi r_2^2} = \frac{4r_1^2}{3r_2^2}$

Substitute $r_1 = \sqrt{3}k$ and $r_2 = 2k$ (using the ratio):

$\frac{4(\sqrt{3}k)^2}{3(2k)^2} = \frac{4 \cdot 3k^2}{3 \cdot 4k^2} = \frac{12k^2}{12k^2} = 1$