If the radius of one sphere is twice the radius of second sphere, then find the ratio of their volumes.

Answer

To find the ratio of the volumes of two spheres where the radius of the first sphere is twice that of the second sphere, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Radii**:
Let the radius of the second sphere be \( r \). According to the problem, the radius of the first sphere \( R \) is twice that of the second sphere. Therefore, we can express this as:
\[ R = 2r \]

2. **Volume Formula for a Sphere**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi R^3 \]
Hence, the volumes of the two spheres can be expressed as:
- Volume of the first sphere \( V_1 \):
\[ V_1 = \frac{4}{3} \pi R^3 \]
- Volume of the second sphere \( V_2 \):
\[ V_2 = \frac{4}{3} \pi r^3 \]

3. **Substituting the Radius of the First Sphere**:
Substitute \( R = 2r \) into the volume formula for the first sphere:
\[ V_1 = \frac{4}{3} \pi (2r)^3 \]

4. **Calculating \( (2r)^3 \)**:
Calculate \( (2r)^3 \):
\[ (2r)^3 = 2^3 \cdot r^3 = 8r^3 \]
Therefore, we can rewrite \( V_1 \):
\[ V_1 = \frac{4}{3} \pi (8r^3) = \frac{32}{3} \pi r^3 \]

5. **Finding the Ratio of the Volumes**:
Now, we can find the ratio of the volumes \( V_1 \) to \( V_2 \):
\[ \frac{V_1}{V_2} = \frac{\frac{32}{3} \pi r^3}{\frac{4}{3} \pi r^3} \]

6. **Simplifying the Ratio**:
Cancel out \( \frac{4}{3} \pi r^3 \) from both the numerator and the denominator:
\[ \frac{V_1}{V_2} = \frac{32}{4} = 8 \]

7. **Final Ratio**:
Thus, the ratio of the volumes of the two spheres is:
\[ V_1 : V_2 = 8 : 1 \]

### Final Answer:
The ratio of the volumes of the two spheres is \( 8 : 1 \).