A sphere and a hemisphere have the same volume. The ratio of their radii is

Options

A. `1:2`
B. `1:8`
C. `1:sqrt(2)`
D. `1:3sqrt(2)`

Answer

To solve the problem of finding the ratio of the radii of a sphere and a hemisphere that have the same volume, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Radii**:
Let the radius of the sphere be \( R \) and the radius of the hemisphere be \( r \).

2. **Volume of the Sphere**:
The volume \( V_s \) of a sphere is given by the formula:
\[ V_s = \frac{4}{3} \pi R^3 \]

3. **Volume of the Hemisphere**:
The volume \( V_h \) of a hemisphere is given by the formula:
\[ V_h = \frac{2}{3} \pi r^3 \]

4. **Set the Volumes Equal**:
Since the volumes of the sphere and the hemisphere are equal, we can set the two volume equations equal to each other:
\[ \frac{4}{3} \pi R^3 = \frac{2}{3} \pi r^3 \]

5. **Cancel Common Terms**:
We can cancel \( \frac{2}{3} \pi \) from both sides:
\[ 2 R^3 = r^3 \]

6. **Express \( r^3 \) in Terms of \( R^3 \)**:
Rearranging gives us:
\[ r^3 = 2 R^3 \]

7. **Take the Cube Root**:
To find the ratio of the radii, we take the cube root of both sides:
\[ r = \sqrt[3]{2} R \]

8. **Find the Ratio**:
The ratio of the radii \( \frac{R}{r} \) can be expressed as:
\[ \frac{R}{r} = \frac{R}{\sqrt[3]{2} R} = \frac{1}{\sqrt[3]{2}} \]

9. **Final Ratio**:
Therefore, the ratio of the radii \( R : r \) is:
\[ R : r = 1 : \sqrt[3]{2} \]

### Conclusion:
The ratio of the radii of the sphere to the hemisphere is \( 1 : \sqrt[3]{2} \).