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 formulas relating the volume \( V \) and surface area \( A \) of a sphere to its radius \( r \) are: \[ V = \frac{4}{3} \pi r^3, \quad A = 4 \pi r^2 \] The volume ratio of two spheres is: \[ \frac{V_1}{V_2} = \left( \frac{r_1}{r_2} \right)^3 \] Given the volume ratio is \( 1:8 \), we get: \[ \left( \frac{r_1}{r_2} \right)^3 = \frac{1}{8} \quad \Rightarrow \quad \frac{r_1}{r_2} = \frac{1}{2} \] The surface area ratio is then: \[ \frac{A_1}{A_2} = \left( \frac{r_1}{r_2} \right)^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \] The answer is \( 1 : 4 \).