If the surface area of a sphere increases by 21%, by what percentage does its volume increase?

Let the original radius of the sphere be r.

Surface area of a sphere is proportional to $r^2$.

If the surface area increases by 21%,

$new\ surface\ area = 1.21 \times original\ surface\ area$

So,

$r_{\text{new}} = \sqrt{1.21},r = 1.1r$

Thus, the radius increases by 10%.

Volume of a sphere is proportional to $r^3$.

Hence,

$Volume\ factor = (1.1)^3 = 1.331$

Percentage increase in volume

$= (1.331 - 1)\times 100 = 33.1%$