Question

The volume of two sphere are in the ratio 27.64. Find their radii if the sum of their radii is 28 cm.

Answer 1

Let the volumes of the spheres be V₁ cm³ and V₂ cm³.
Also,
Let the radii of the spheres be r₁ cm and r₂ cm.
Now, according to the question, we have:
$\frac{V_1}{V_2}=\frac{72}{64}$ …(i)
We know:
Volume of a sphere = $\frac{4}{3}\mathrm{\pi }{r}^{\mathit{3}}$
From (i), we get:
$\frac{\frac{4}{3}\mathrm{\pi }{r_1}^{\mathit{3}}}{\frac{4}{3}\mathrm{\pi }{r_2}^{\mathit{3}}}=\frac{27}{64}$
$$\begin{gathered} \Rightarrow {\left({\displaystyle \frac{r_1}{r_2}}\right)}^3=\frac{27}{64} \\ \Rightarrow {\left({\displaystyle \frac{r_1}{r_2}}\right)}^3={\left(\frac{3}{4}\right)}^3 \\ \Rightarrow \frac{r_1}{r_2}=\frac{3}{4} \end{gathered}$$

$\Rightarrow r_1=\frac{3}{4}{r}_2$
Also,
r₁ + r₂ = 28 (Given)
$$\begin{gathered} \Rightarrow \frac{3}{4}{r}_2+r_2=28 \\ \Rightarrow \frac{7}{4}{r}_2=28 \\ \Rightarrow r_2=28\times \frac{4}{7} \\ \Rightarrow r_2=16\mathrm{cm} \\ \Rightarrow r_1=16\times \frac{3}{4} \\ =12\mathrm{cm} \end{gathered}$$

Hence, the radii of the spheres are 12 cm and 16 cm.