If the surface areas of two spheres are in the ratio 4 : 25, then the ratio of their volumes is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 8 : 125
The surface area of a sphere is \(4\pi r^2\) and its volume \(\frac{4}{3}\pi r^3\). So the ratio of surface areas is the square of the ratio of radii, and the ratio of volumes is its cube.
The surface areas give the ratio of the radii: \(\displaystyle \frac{r_1^2}{r_2^2}=\frac{4}{25}\), so \(\displaystyle \frac{r_1}{r_2}=\frac{2}{5}\)
Ratio of the volumes: \(\displaystyle \frac{V_1}{V_2}=\left(\frac{2}{5}\right)^3=\frac{8}{125}\)