Question

The radius of Jupiter is $11$ times the radius of the Earth. Calculate the ratio of the volumes of Jupiter and the Earth. How many Earths can Jupiter accommodate?

Answer 1

Step-1: Ratio of the volumes of Jupiter and the Earth:

Let the radius of the earth be $r$.

Then, the radius of the Jupiter is $11r$.

Volume of the sphere$\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}\mathbf{\pi }\mathbf{\times }{\mathbf{r}}^{\mathbf{3}}$, where $r$ is the radius of the sphere.

Now,

Volume of the Earth$\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}\mathbf{\pi }\mathbf{\times }{\mathbf{r}}^{\mathbf{3}}$

Volume of the Jupiter$\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}\mathbf{\pi }\mathbf{\times }{\left(11r\right)}^{\mathbf{3}}$

Required ratio

$$\begin{gathered} =\frac{\mathrm{Volume}\mathrm{of}\mathrm{Jupiter}}{\mathrm{Volume}\mathrm{of}\mathrm{Earth}} \\ =\frac{{\displaystyle \frac{4}{3}}\mathrm{\pi }\times {\left(11r\right)}^3}{\frac{4}{3}\mathrm{\pi }\times r^3} \\ =\frac{{\left(11r\right)}^3}{r^3} \\ =\frac{11r\times 11r\times 11r}{r^3} \\ =\frac{1331r^3}{r^3} \\ =\frac{1331}{1} \end{gathered}$$

The ratio is $1331:1$.

Step-2 : How many earth can jupiter accommodate:

Jupiter can accommodate $1331$ Earths.

Hence, the required ratio is $1331:1$ and Jupiter can accommodate $1331$ Earths.