The error in measurement of radius of a sphere is 0.1% then error in its volume is -

Options

A. `0.3%`
B. `0.4%`
C. `0.5%`
D. `0.6%`

Explanation

To solve the problem of finding the error in the volume of a sphere given the error in the radius, we can follow these steps:

### Step-by-Step Solution:

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

2. **Differentiate the Volume with Respect to Radius**:
To find the relationship between the error in volume and the error in radius, we differentiate the volume with respect to the radius \( r \):
\[ dV = 4 \pi r^2 \, dr \]

3. **Express the Relative Error in Volume**:
We can express the relative error in volume \( \frac{dV}{V} \):
\[ \frac{dV}{V} = \frac{4 \pi r^2 \, dr}{\frac{4}{3} \pi r^3} \]
Simplifying this gives:
\[ \frac{dV}{V} = \frac{3 \, dr}{r} \]

4. **Relate the Errors**:
The relative error in volume can be expressed in terms of the relative error in radius:
\[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \]
where \( \Delta V \) is the absolute error in volume and \( \Delta r \) is the absolute error in radius.

5. **Convert to Percentage Error**:
To find the percentage error in volume, we multiply both sides by 100:
\[ \frac{\Delta V}{V} \times 100 = 3 \left( \frac{\Delta r}{r} \times 100 \right) \]

6. **Substitute the Given Error in Radius**:
We are given that the error in measurement of radius \( \frac{\Delta r}{r} \) is 0.1%. Substituting this value:
\[ \frac{\Delta V}{V} \times 100 = 3 \times 0.1 = 0.3\% \]

7. **Conclusion**:
Therefore, the error in the volume of the sphere is:
\[ \text{Error in Volume} = 0.3\% \]

### Final Answer:
The error in the volume of the sphere is **0.3%**.