The ratio of surface area of two spheres is 4 : 9, the ratio of there volumes is :

Question

Accepted Answer

To determine the ratio of the volumes of two spheres given the ratio of their surface areas, we need to understand the mathematical relationship between the surface area and volume of a sphere.

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

2. **Given Ratio of Surface Areas:**
   - The ratio of the surface areas of two spheres is 4:9. Let's denote the radii of the two spheres as $ r_1 $ and $ r_2 $.
   - Therefore, $\frac{4\pi r_1^2}{4\pi r_2^2} = \frac{4}{9}$.
   - Simplifying, we get $\frac{r_1^2}{r_2^2} = \frac{4}{9}$.

3. **Finding the Ratio of Radii:**
   - Taking the square root of both sides, we find $\frac{r_1}{r_2} = \frac{2}{3}$.

4. **Ratio of Volumes:**
   - Using the volume formula, the ratio of the volumes of the two spheres is $\frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} = \frac{r_1^3}{r_2^3}$.
   - Substituting the ratio of the radii, we get $\left(\frac{2}{3}\right)^3 = \frac{8}{27}$.

Therefore, the ratio of the volumes of the two spheres is 8:27, which corresponds to Option 3.

**Explanation of Incorrect Options:**

- **Option 1 (2:3):** This option suggests a direct ratio of radii, not volumes. The ratio of the radii is indeed 2:3, but when cubed to find the volume ratio, it becomes 8:27.

- **Option 2 (4:9):** This is the ratio of the surface areas, not the volumes. The volume ratio is derived from the cube of the radius ratio, not the square.

- **Option 4 (64:729):** This would be the result of cubing a different ratio, specifically $\left(\frac{4}{9}\right)^{3/2}$, which is not applicable here as it does not represent the correct transformation from surface area to volume.

Thus, Option 3 (8:27) is the correct answer, as it accurately reflects the relationship between the surface area and volume ratios of the spheres.