A golf ball has diameter equal to 4.1 cm. Its surface has 150 dimples each of…

When n identical hemispherical dimples of radius r are carved into a sphere's surface, the total exposed surface area equals the sphere's own surface area plus the net area gained per dimple, summed over all dimples. Carving a hemispherical dimple removes a flat circular disc of area πr2 from the sphere's surface but exposes the larger curved hemispherical surface of area 2πr2 in its place, so each dimple contributes a net gain of πr2 (= 2πr2 − πr2).

1. Find the sphere's radius: R = diameter ÷ 2 = 4.1 ÷ 2 = 2.05 cm.

2. Compute the sphere's own surface area: 4πR2 = 4π(2.05)2 = 4π(4.2025) = 16.81π cm2.

3. Compute the net area gained per dimple: πr2 = π(0.2)2 = 0.04π cm2; over n = 150 dimples that is 150 × 0.04π = 6π cm2.

4. Add the two contributions: Total = 16.81π + 6π = 22.81π cm2.

5. Substitute π ≈ 3.14: Total = 22.81 × 3.14 = 71.6234 ≈ 71.62 cm2.

As an independent check, the sphere's own surface area alone (ignoring the dimples) is 16.81π ≈ 52.78 cm2. Adding the net gain of 6π ≈ 18.84 cm2 from the 150 dimples gives 52.78 + 18.84 = 71.62 cm2 — the same total, confirming the net-gain-per-dimple reasoning is consistent.