Question

The surface areas of two spheres are in the ratio$16:9$. The ratio of their volumes is :

• A. $64:27$
• B. $16:9$
• C. $4:3$
• D. ${16}^3:9^3$

Answer 1

The correct option is A

$64:27$

Step 1: Finding the ratio of radii

Let the radius be $\text{'}{\mathrm{r}}_1\text{'}$ and $\text{'}{r}_2\text{'}$ of two spheres.

Also,

Surface area of two sphere is in ratio $16:9$

$$\begin{gathered} \Rightarrow \frac{4{{\mathrm{\pi r}}_1}^2}{4{{\mathrm{\pi r}}_1}^2}=\frac{16}{9} \\ \Rightarrow \frac{{r_1}^2}{{r_2}^2}=\frac{16}{9} \\ \Rightarrow \frac{r_1}{r_2}=\frac{4}{3}......\left(1\right) \end{gathered}$$

Step 2: Finding the ratio of volumes

Ratio of volume of spheres $=\frac{4}{3}{{\mathrm{\pi r}}_1}^3:\frac{4}{3}{{\mathrm{\pi r}}_2}^3$

$={\left(\frac{r_1}{r_2}\right)}^3$

$$\begin{gathered} ={\left(\frac{4}{3}\right)}^3[U\sin g(1\left)\right] \\ =\frac{64}{27} \end{gathered}$$

Final answer: Hence, the correct option is option A.