The measurement of radius of a circle has error of 1%. The error in measurement of its area

Options

A. 0.01
B. 0.02
C. 0.03
D. none of these

Answer

To solve the problem of determining the error in the measurement of the area of a circle when the radius has a 1% error, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We know that the radius \( r \) of a circle has an error of 1%. We need to find the error in the area \( A \) of the circle.

2. **Formula for Area of a Circle**: The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]

3. **Determine the Relative Error in Radius**: The relative error in the measurement of the radius is given as:
\[ \frac{\Delta r}{r} = 0.01 \quad \text{(which is 1%)} \]
Here, \( \Delta r \) is the absolute error in the radius.

4. **Use the Formula for Area Error**: The formula for the relative error in the area \( A \) is related to the relative error in the radius. Specifically, we have:
\[ \frac{\Delta A}{A} = 2 \frac{\Delta r}{r} \]
This means that the error in the area is twice the error in the radius.

5. **Substitute the Known Values**: Now substituting the value of the relative error in the radius into the equation for the area:
\[ \frac{\Delta A}{A} = 2 \times 0.01 = 0.02 \]

6. **Conclusion**: The relative error in the area is 0.02, which means the area has a 2% error.

### Final Answer:
The error in the measurement of the area of the circle is **0.02**.