Consider the ratio of the surface area of a sphere to its volume. What happens to this ratio if the diameter of the sphere increases by a factor of 3?
A) increases by a factor of 3
B) decreases by a factor of 3
C) increases by a factor of 9
D) decreases by a factor of 9
E) stays the same

Question

Quizplus · Accepted Answer

The surface area of a sphere is given by $4\pi r^2$, and its volume is given by $\frac{4}{3}\pi r^3$, where $r$ is the radius. The ratio of surface area to volume is therefore $\frac{4\pi r^2}{\frac{4}{3}\pi r^3} = \frac{3}{r}$. If the diameter (and thus the radius) increases by a factor of 3, the ratio decreases by a factor of 3, because the ratio is inversely proportional to the radius.

Consider the Ratio of the Surface Area of a Sphere