If the surface areas of two spheres are in the ratio 4:9 , then the ratio of their volumes is :

Options

A. `8:25`
B. `8:26`
C. `8:27`
D. `8:28`

Answer

To solve the problem of finding the ratio of the volumes of two spheres given the ratio of their surface areas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Ratio of Surface Areas**:
The surface areas of the two spheres are given in the ratio 4:9. We can express this mathematically as:
\[ \frac{S_1}{S_2} = \frac{4}{9} \]

2. **Use the Formula for Surface Area of a Sphere**:
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4\pi r^2 \]
where \( r \) is the radius of the sphere. For our two spheres, we can write:
\[ S_1 = 4\pi r_1^2 \quad \text{and} \quad S_2 = 4\pi r_2^2 \]

3. **Set Up the Equation Using the Surface Area Ratio**:
Substituting the expressions for \( S_1 \) and \( S_2 \) into the ratio gives us:
\[ \frac{4\pi r_1^2}{4\pi r_2^2} = \frac{4}{9} \]
The \( 4\pi \) cancels out, leading to:
\[ \frac{r_1^2}{r_2^2} = \frac{4}{9} \]

4. **Take the Square Root to Find the Ratio of Radii**:
Taking the square root of both sides, we find:
\[ \frac{r_1}{r_2} = \sqrt{\frac{4}{9}} = \frac{2}{3} \]

5. **Find the Ratio of Volumes**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]
For our two spheres, we can write:
\[ V_1 = \frac{4}{3}\pi r_1^3 \quad \text{and} \quad V_2 = \frac{4}{3}\pi r_2^3 \]
The ratio of their volumes is:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} = \frac{r_1^3}{r_2^3} \]

6. **Substitute the Ratio of Radii**:
We already found that \( \frac{r_1}{r_2} = \frac{2}{3} \). Thus:
\[ \frac{V_1}{V_2} = \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} \]

7. **Conclusion**:
Therefore, the ratio of the volumes of the two spheres is:
\[ \frac{V_1}{V_2} = \frac{8}{27} \]

### Final Answer:
The ratio of their volumes is \( 8:27 \).