If the ratio of the volume of two spheres is 1 : 8, then the ratio of their surface area is :

Question

Accepted Answer

To determine the ratio of the surface areas of two spheres given that the ratio of their volumes is 1:8, we can use the mathematical relationships between the volume and surface area of spheres.

1. **Volume and Surface Area Formulas:**
   - The volume $ V $ of a sphere is given by the formula: $ V = \frac{4}{3} \pi r^3 $, where $ r $ is the radius of the sphere.
   - The surface area $ A $ of a sphere is given by the formula: $ A = 4 \pi r^2 $.

2. **Given Ratio of Volumes:**
   - The ratio of the volumes of the two spheres is 1:8. This implies:
     $$
     \frac{V_1}{V_2} = \frac{1}{8}
     $$
   - Using the volume formula, we have:
     $$
     \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \frac{1}{8}
     $$
   - Simplifying, we get:
     $$
     \left(\frac{r_1}{r_2}\right)^3 = \frac{1}{8}
     $$
   - Taking the cube root of both sides, we find:
     $$
     \frac{r_1}{r_2} = \frac{1}{2}
     $$

3. **Finding the Ratio of Surface Areas:**
   - Using the surface area formula, the ratio of the surface areas is:
     $$
     \frac{A_1}{A_2} = \frac{4 \pi r_1^2}{4 \pi r_2^2} = \left(\frac{r_1}{r_2}\right)^2
     $$
   - Substituting the ratio of the radii:
     $$
     \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
     $$

Therefore, the ratio of the surface areas of the two spheres is 1:4, which corresponds to Option 2.

**Explanation of Incorrect Options:**

- **Option 1 (1:2):** This option is incorrect because it suggests that the surface area ratio is the square of the radius ratio, which is not the case. The correct calculation shows that the surface area ratio is 1:4, not 1:2.

- **Option 3 (1:8):** This option is incorrect because it confuses the ratio of volumes with the ratio of surface areas. The surface area ratio is determined by the square of the radius ratio, not the cube.

- **Option 4 (1:16):** This option is incorrect because it suggests that the surface area ratio is the fourth power of the radius ratio, which is not applicable. The correct surface area ratio is the square of the radius ratio, resulting in 1:4.

In conclusion, Option 2 is the correct answer because it accurately reflects the mathematical relationship between the volumes and surface areas of spheres.