If the surface areas of two spheres are in the ratio 9:16, the ratio of their volumes is

Options

A. `16:9`
B. `27:64`
C. `64:27`
D. `9:16`

Answer

To find 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 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.

2. **Set up the ratio of surface areas**:
Let the surface areas of the two spheres be \( S_1 \) and \( S_2 \). According to the problem, the ratio of their surface areas is given as:
\[ \frac{S_1}{S_2} = \frac{9}{16} \]

3. **Express the surface areas in terms of their radii**:
Using the formula for surface area, we can express the ratio in terms of the radii \( r_1 \) and \( r_2 \):
\[ \frac{4 \pi r_1^2}{4 \pi r_2^2} = \frac{9}{16} \]
The \( 4 \pi \) cancels out:
\[ \frac{r_1^2}{r_2^2} = \frac{9}{16} \]

4. **Take the square root of both sides**:
To find the ratio of the radii, we take the square root of both sides:
\[ \frac{r_1}{r_2} = \frac{3}{4} \]

5. **Understand the formula for volume of a sphere**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

6. **Set up the ratio of volumes**:
The ratio of the volumes \( V_1 \) and \( V_2 \) of the two spheres can be expressed as:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} \]
Again, the \( \frac{4}{3} \pi \) cancels out:
\[ \frac{V_1}{V_2} = \frac{r_1^3}{r_2^3} \]

7. **Substitute the ratio of the radii**:
Now, substituting the ratio of the radii we found earlier:
\[ \frac{V_1}{V_2} = \frac{(3)^3}{(4)^3} = \frac{27}{64} \]

8. **Conclusion**:
Thus, the ratio of the volumes of the two spheres is:
\[ \frac{V_1}{V_2} = \frac{27}{64} \]

### Final Answer:
The ratio of their volumes is \( 27:64 \).