If the radius of a sphere is increased by 50 percent, then how much percentage…

For a sphere, the total surface area is A = 4πr2, so the surface area is directly proportional to the square of the radius: A ∝ r2. This means that if the radius is scaled by a factor k, the area is scaled by k2.

1. The radius is increased by 50%, so the new radius is 1.5 times the original radius: r' = 1.5r, i.e. the scale factor k = 1.5.

2. Since A ∝ r2, the new area is A' = k2 × A = (1.5)2 × A = 2.25A.

3. The increase in area is A' − A = 2.25A − A = 1.25A, which is a 125% increase over the original area.

Cross-check using the percentage-change identity for a squared quantity: if a quantity increases by x%, its square increases by (2x + x2/100)%. Here x = 50, so the increase is 2(50) + 502/100 = 100 + 25 = 125%, confirming the result above.