Question

The ratio of volumes of two spheres is \( 125 : 27 \). The ratio of their surface areas is:

• A. \( 9 : 25 \)
• B. \( 25 : 9 \)
• C. \( 5 : 3 \)
• D. \( 3 : 5 \)

Hint:

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

Answer 1

The Correct Option is B

The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3, \] and the surface area \( A \) is given by the formula: \[ A = 4 \pi r^2. \] Let the radii of the two spheres be \( r_1 \) and \( r_2 \), and the ratio of their volumes is given as: \[ \frac{V_1}{V_2} = \frac{125}{27}. \] Using the volume formula: \[ \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \frac{125}{27} \quad \Rightarrow \quad \frac{r_1^3}{r_2^3} = \frac{125}{27}. \] Taking the cube root of both sides: \[ \frac{r_1}{r_2} = \frac{5}{3}. \] Now, the ratio of the surface areas is: \[ \frac{A_1}{A_2} = \frac{4 \pi r_1^2}{4 \pi r_2^2} = \frac{r_1^2}{r_2^2} = \left( \frac{5}{3} \right)^2 = \frac{25}{9}. \] Thus, the ratio of their surface areas is \( \boxed{25 : 9} \).