If the ratio of the radii of the sphere and hemisphere is √3:2 , then…

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

The total surface area of a sphere is \(4\pi r^2\). A solid hemisphere has its curved surface \(2\pi R^2\) plus the flat circular base \(\pi R^2\), so its total surface area is \(3\pi R^2\).

Taking the radii (in the ratio \(\sqrt{3}:2\)):

• Sphere: \(r=\sqrt{3}k\)

• Hemisphere: \(R=2k\)

The two surface areas (putting \(r=\sqrt{3}k\) in \(4\pi r^2\) and \(R=2k\) in \(3\pi R^2\)):

• Sphere: \(4\pi\times(\sqrt{3}k)^2=4\pi\times3k^2=12\pi k^2\)

• Hemisphere: \(3\pi\times(2k)^2=3\pi\times4k^2=12\pi k^2\)

• The two areas are equal, so the ratio is 1:1