Question

The volumes of the two spheres are in a ratio of $1:8$.

What is the ratio of their radii?

Answer 1

Determine the ratio of their radii:

The volume of the sphere $V$ is calculated by $V=\frac{4}{3}{\mathrm{\pi r}}^3$ where $r$ represents the radius of the sphere.

Let us assume $r_1$ and $r_2$ be the radius of the first and second sphere respectively.

The volume of the first sphere $V_1$ is $\frac{4}{3}{\mathrm{\pi r}}_1^3$.

The volume of the second sphere $V_2$ is $\frac{4}{3}{\mathrm{\pi r}}_2^3$.

It is given that the volumes of the two spheres are in a ratio of $1:8$.

$\begin{array}{rcl}{V}_1:V_2& =& 1:8\\ & \Rightarrow & \frac{4}{3}{\mathrm{\pi r}}_1^3:\frac{4}{3}{\mathrm{\pi r}}_2^3=1:8\\ & \Rightarrow & {\mathrm{r}}_1^3:{\mathrm{r}}_2^3=1:8\end{array}$

Take the cube root on both sides.

$\begin{array}{rcl}& \Rightarrow & \sqrt[3]{{\mathrm{r}}_1^3}:\sqrt[3]{{\mathrm{r}}_2^3}=\sqrt[3]{1}:\sqrt[3]{8}\\ & \Rightarrow & r_1:r_2=1:\sqrt[3]{2^3}\\ & \Rightarrow & =1:2\end{array}$

Hence, the ratio of their radii is $\mathbf{1}\mathbf{:}\mathbf{2}$.