Question

If the ratio of volumes of two spheres is 1 : 8, then the ratio of their surface areas is:

• A. 1 : 6
• B. 1 : 2
• C. 1 : 4
• D. 1 : 8

Hint:

For spheres, the ratio of the surface areas is the square of the ratio of their radii, and the ratio of their volumes is the cube of the ratio of their radii.

Answer 1

The Correct Option is A

The volume \( V \) and surface area \( A \) of a sphere are related to its radius \( r \) by the following formulas: \[ V = \frac{4}{3} \pi r^3, \quad A = 4 \pi r^2 \] The ratio of the volumes of two spheres is given by: \[ \frac{V_1}{V_2} = \left( \frac{r_1}{r_2} \right)^3 \] Since the ratio of the volumes is \( 1:8 \), we have: \[ \left( \frac{r_1}{r_2} \right)^3 = \frac{1}{8} \quad \Rightarrow \quad \frac{r_1}{r_2} = \frac{1}{2} \] Now, the ratio of the surface areas is: \[ \frac{A_1}{A_2} = \left( \frac{r_1}{r_2} \right)^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \] Thus, the correct answer is \( 1 : 4 \).