A sphere and a hemisphere have the same volume. The ratio of their curved surface area is :

Options

A. `2^(-(3)/(2)) :1`
B. `2^((2)/(3)) :1`
C. `4^(-(2)/(3)) :1`
D. `2^((1)/(3)) :1`

Answer

To solve the problem of finding the ratio of the curved surface areas 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 also 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 Volumes Equal**:
Since the volumes of the sphere and hemisphere are equal, we can set the two 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:
\[ 2R^3 = r^3 \]

6. **Solve for r**:
Taking the cube root of both sides gives:
\[ r = R \left( \frac{1}{2} \right)^{1/3} \]

7. **Curved Surface Area of the Sphere**:
The curved surface area \( A_s \) of the sphere is given by:
\[ A_s = 4 \pi R^2 \]

8. **Curved Surface Area of the Hemisphere**:
The curved surface area \( A_h \) of the hemisphere is given by:
\[ A_h = 2 \pi r^2 \]

9. **Substituting r**:
Substitute \( r \) from step 6 into the hemisphere's surface area formula:
\[ A_h = 2 \pi \left( R \left( \frac{1}{2} \right)^{1/3} \right)^2 = 2 \pi R^2 \left( \frac{1}{2} \right)^{2/3} \]

10. **Calculate the Ratio**:
Now, we can find the ratio of the curved surface areas:
\[ \text{Ratio} = \frac{A_s}{A_h} = \frac{4 \pi R^2}{2 \pi R^2 \left( \frac{1}{2} \right)^{2/3}} = \frac{4}{2 \left( \frac{1}{2} \right)^{2/3}} = \frac{4}{2} \cdot \left( 2^{2/3} \right) = 2 \cdot 2^{2/3} \]

11. **Final Simplification**:
The ratio simplifies to:
\[ \text{Ratio} = 2^{1 + 2/3} = 2^{5/3} \]

### Final Answer:
The ratio of the curved surface area of the sphere to the curved surface area of the hemisphere is \( 2^{5/3} \).