The relative error in the determination of the surface area of a sphere is `alpha`. Then the relative error in the determination of its volume is :

Options

A. `3/2 alpha`
B. `2/3 alpha`
C. `alpha`
D. `5/2 alpha`

Answer

To solve the problem, we need to find the relative error in the volume of a sphere given the relative error in the surface area of the sphere is denoted as \( \alpha \).

### Step-by-Step Solution:

1. **Understand the Surface Area and Volume Formulas:**
- The surface area \( S \) of a sphere is given by:
\[ S = 4 \pi r^2 \]
- The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]

2. **Determine the Relative Error in Surface Area:**
- The relative error in the surface area is given as:
\[ \frac{\Delta S}{S} = \alpha \]
- Here, \( \Delta S \) is the absolute error in the surface area.

3. **Differentiate the Surface Area Formula:**
- Taking the natural logarithm of the surface area:
\[ \ln S = \ln(4 \pi) + \ln(r^2) \]
- Differentiating both sides gives:
\[ \frac{1}{S} \Delta S = 2 \frac{1}{r} \Delta r \]
- Rearranging this, we find:
\[ \frac{\Delta S}{S} = 2 \frac{\Delta r}{r} \]

4. **Relate the Relative Error in Radius:**
- From the equation \( \frac{\Delta S}{S} = \alpha \), we can substitute:
\[ \alpha = 2 \frac{\Delta r}{r} \]
- Therefore, we can express the relative error in radius as:
\[ \frac{\Delta r}{r} = \frac{\alpha}{2} \]

5. **Determine the Relative Error in Volume:**
- Now, we need to find the relative error in the volume:
\[ \frac{\Delta V}{V} \]
- Taking the natural logarithm of the volume:
\[ \ln V = \ln\left(\frac{4}{3} \pi\right) + 3 \ln(r) \]
- Differentiating both sides gives:
\[ \frac{1}{V} \Delta V = 3 \frac{1}{r} \Delta r \]
- Rearranging this, we find:
\[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \]

6. **Substituting the Relative Error in Radius:**
- Now substituting \( \frac{\Delta r}{r} = \frac{\alpha}{2} \) into the volume equation:
\[ \frac{\Delta V}{V} = 3 \left(\frac{\alpha}{2}\right) = \frac{3}{2} \alpha \]

### Conclusion:
The relative error in the determination of the volume of the sphere is:
\[ \frac{\Delta V}{V} = \frac{3}{2} \alpha \]

### Final Answer:
The correct option is \( \frac{3}{2} \alpha \).