If error in radius is 3% whast is error in volume of sphere?

Options

A. 0.03
B. 0.27
C. 0.09
D. 0.06

Explanation

To find the error in the volume of a sphere when the error in the radius is given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between volume and radius**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
This indicates that the volume is proportional to the cube of the radius.

2. **Identify the percentage error in radius**:
We are given that the error in the radius \( r \) is 3%. This can be expressed as:
\[ \frac{\Delta r}{r} \times 100 = 3\% \]
where \( \Delta r \) is the absolute error in the radius.

3. **Determine the relationship between volume error and radius error**:
Since \( V \) is proportional to \( r^3 \), we can express the error in volume \( \Delta V \) in terms of the error in radius \( \Delta r \):
\[ \frac{\Delta V}{V} = 3 \cdot \frac{\Delta r}{r} \]
This means that the percentage error in volume is three times the percentage error in radius.

4. **Substituting the known error**:
Now, substituting the known error in radius into the equation:
\[ \frac{\Delta V}{V} = 3 \cdot \frac{3}{100} = \frac{9}{100} \]

5. **Convert to percentage**:
To express this as a percentage:
\[ \frac{\Delta V}{V} \times 100 = 9\% \]

### Final Answer:
The error in the volume of the sphere is **9%**.