If the radius of a sphere is doubled, what is the ratio of the volumes of the first sphere to that of the second sphere?

The radius of a sphere is doubled.

To do:

We have to find the ratio of the volumes of the first sphere to that of the second sphere.

Solution:

Let $r$ be the radius of the given sphere.

This implies,
Volume of the sphere $=\frac{4}{3} \pi r^3$

The radius of the new sphere $= 2r$

Therefore,

Volume of the new sphere $=\frac{4}{3} \pi(2 r)^{3}$

$=\frac{4}{3} \pi \times 8 r^{3}$

$=8(\frac{4}{3} \pi r^{3})$

Ratio of the volume of the original sphere and the new sphere $=\frac{4}{3} \pi r^{3}: 8 \times \frac{4}{3} \pi r^{3}$

$=1: 8$