Question

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

Answer 1

Find the ratio of the volume of the first sphere to that of the second sphere.

Let $V_1$ be the volume of first sphere and $V_2$ be the volume of second sphere.

Also, $r_2=2r_1$ (Where $r_1$ is the radius of first sphere and $r_2$ be the radius of second sphere)

The volume of a sphere of radius $r$ $=\frac{4}{3}{\mathrm{\pi r}}^3$

$$\begin{gathered} \therefore \frac{V_1}{V_2}=\frac{{\displaystyle \frac{4}{3}}{{\mathrm{\pi r}}_1}^3}{{\displaystyle \frac{4}{3}}{{\mathrm{\pi r}}_2}^3} \\ \Rightarrow \frac{V_1}{V_2}=\frac{{\displaystyle \frac{4}{3}{{\mathrm{\pi r}}_1}^3}}{{\displaystyle \frac{4}{3}}{\mathrm{\pi }\left(2r_1\right)}^3} \\ \Rightarrow \frac{V_1}{V_2}=\frac{1}{8} \end{gathered}$$

Hence, the required ratio is $\frac{1}{8}$.