Volume of two spheres is in the ratio 64 : 125. Find the ratio of their surface areas.

Answer

To solve the problem of finding the ratio of the surface areas of two spheres given that their volumes are in the ratio 64:125, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Ratio of Volumes**:
We are given that the volumes of two spheres are in the ratio:
\[ V_1 : V_2 = 64 : 125 \]
This can be expressed as:
\[ \frac{V_1}{V_2} = \frac{64}{125} \]

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

3. **Express the Volumes in Terms of Radii**:
Let the radii of the two spheres be \( r_1 \) and \( r_2 \). Therefore, we can write:
\[ V_1 = \frac{4}{3} \pi r_1^3 \]
\[ V_2 = \frac{4}{3} \pi r_2^3 \]

4. **Set Up the Equation Using the Volume Ratio**:
Substituting the expressions for \( V_1 \) and \( V_2 \) into the volume ratio gives:
\[ \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \frac{64}{125} \]
The \( \frac{4}{3} \pi \) cancels out, leading to:
\[ \frac{r_1^3}{r_2^3} = \frac{64}{125} \]

5. **Find the Ratio of Radii**:
To find the ratio of the radii, take the cube root of both sides:
\[ \frac{r_1}{r_2} = \sqrt[3]{\frac{64}{125}} = \frac{\sqrt[3]{64}}{\sqrt[3]{125}} = \frac{4}{5} \]

6. **Recall the Formula for Surface Area of a Sphere**:
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4 \pi r^2 \]

7. **Express the Surface Areas in Terms of Radii**:
Let \( S_1 \) and \( S_2 \) be the surface areas of the two spheres:
\[ S_1 = 4 \pi r_1^2 \]
\[ S_2 = 4 \pi r_2^2 \]

8. **Set Up the Equation Using the Surface Area Ratio**:
The ratio of the surface areas can be expressed as:
\[ \frac{S_1}{S_2} = \frac{4 \pi r_1^2}{4 \pi r_2^2} = \frac{r_1^2}{r_2^2} \]

9. **Substitute the Ratio of Radii**:
We already found that:
\[ \frac{r_1}{r_2} = \frac{4}{5} \]
Therefore:
\[ \frac{S_1}{S_2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]

10. **Final Answer**:
The ratio of the surface areas of the two spheres is:
\[ S_1 : S_2 = 16 : 25 \]