Question

If ratio of volumes of two spheres is \( 27 : 64 \), then ratio of their surface areas will be:

• A. 9:16
• B. 16:9
• C. 3:4
• D. 4:3

Hint:

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

Answer 1

The Correct Option is A

We are given that the ratio of the volumes of two spheres is \( 27 : 64 \). The formulas for the volume and surface area of a sphere are: \[ V = \frac{4}{3} \pi r^3 \quad \text{(Volume of sphere)} \] \[ A = 4 \pi r^2 \quad \text{(Surface area of sphere)} \]
Step 1: Relating the ratio of volumes to the ratio of radii.
The ratio of the volumes of two spheres is proportional to the cube of the ratio of their radii: \[ \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3 \] Given \( \frac{V_1}{V_2} = \frac{27}{64} \), we find the ratio of the radii: \[ \left(\frac{r_1}{r_2}\right)^3 = \frac{27}{64} \] \[ \frac{r_1}{r_2} = \frac{3}{4} \]
Step 2: Relating the ratio of surface areas to the ratio of radii.
The ratio of the surface areas of the spheres is proportional to the square of the ratio of their radii: \[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 \] Substituting the value \( \frac{r_1}{r_2} = \frac{3}{4} \): \[ \frac{A_1}{A_2} = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]
Step 3: Conclusion.
Thus, the ratio of the surface areas of the two spheres is \( 9 : 16 \). The correct answer is (A).