If volume of two spheres are in the ratio `64: 27`, then the ratio of their surface area is

Options

A. `3:4`
B. `4:3`
C. `9:16`
D. `16:9`

Explanation

To solve the problem of finding the ratio of the surface areas of two spheres given that their volumes are in the ratio of 64:27, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume Ratio**:
We are given that the volumes of two spheres are in the ratio \( V_1 : V_2 = 64 : 27 \).

2. **Volume Formula**:
The formula for the volume of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

3. **Set Up the Equation**:
Let the radius of the first sphere be \( R_1 \) and the radius of the second sphere be \( R_2 \). Therefore, we can write:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi R_1^3}{\frac{4}{3} \pi R_2^3} = \frac{R_1^3}{R_2^3} \]
Given that \( \frac{V_1}{V_2} = \frac{64}{27} \), we have:
\[ \frac{R_1^3}{R_2^3} = \frac{64}{27} \]

4. **Find the Ratio of Radii**:
Taking the cube root of both sides gives us:
\[ \frac{R_1}{R_2} = \frac{\sqrt[3]{64}}{\sqrt[3]{27}} = \frac{4}{3} \]

5. **Surface Area Formula**:
The formula for the surface area of a sphere is:
\[ S = 4 \pi r^2 \]
Therefore, the surface areas of the two spheres can be expressed as:
\[ S_1 = 4 \pi R_1^2 \quad \text{and} \quad S_2 = 4 \pi R_2^2 \]

6. **Set Up the Surface Area Ratio**:
The ratio of the surface areas is:
\[ \frac{S_1}{S_2} = \frac{4 \pi R_1^2}{4 \pi R_2^2} = \frac{R_1^2}{R_2^2} \]

7. **Substitute the Radius Ratio**:
We already found that \( \frac{R_1}{R_2} = \frac{4}{3} \). Therefore:
\[ \frac{S_1}{S_2} = \left(\frac{R_1}{R_2}\right)^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]

8. **Conclusion**:
The ratio of the surface areas of the two spheres is:
\[ S_1 : S_2 = 16 : 9 \]