A golf ball has a diameter equal to $4.2cm.$ Its surface has $200$ dimples each of radius $2mm.$ Assuming that the dimples are hemispherical, the total surface area which is exposed to the surroundings is

The correct option is D: $80.58c{m}^2$

Radius of the ball $=\frac{4.2}{2}=2.1cm$

Surface area of the ball $=4\pi r^2=4\times \pi (\frac{4.2}{2}{)}^2=17.64\pi c{m}^2$

In case of each dimple, surface area is equal to $\pi r^2$ is removed from the surface of the ball where as the surface area of hemisphere $=2\pi r^2$ is exposed to the surroundings.

$\therefore$ Total surface exposed to the surrounding = Surface area of the ball $+200\times \pi r^2$

$=17.64\pi +200\pi \times (\frac{2}{10}{)}^2$ [$\because$ Radius of ball $=2mm=\frac{2}{10}cm$]

$=17.64\pi +8\pi$

$=25.64\times \frac{22}{7}c{m}^2$

$=\frac{564.08}{7}=80.58c{m}^2$

Hence, the total surface area exposed to the surroundings is $80.58c{m}^2$