If the error in the measurement of radius of circle is 3% then calculate the error in measurement of area of circle :-

Options

A. `5%`
B. `8%`
C. `3%`
D. `6%`

Answer

To solve the problem of calculating the error in the measurement of the area of a circle when the error in the measurement of the radius is given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between radius and area**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

2. **Differentiate the area with respect to the radius**:
To find the error in the area, we can differentiate the area with respect to the radius:
\[ dA = \frac{dA}{dr} \cdot dr \]
The derivative of \( A \) with respect to \( r \) is:
\[ \frac{dA}{dr} = 2\pi r \]
Thus, we have:
\[ dA = 2\pi r \cdot dr \]

3. **Express the relative error in area**:
The relative error in the area can be expressed as:
\[ \frac{dA}{A} = \frac{dA}{\pi r^2} \]
Substituting \( dA \) from the previous step:
\[ \frac{dA}{A} = \frac{2\pi r \cdot dr}{\pi r^2} \]
Simplifying this gives:
\[ \frac{dA}{A} = \frac{2dr}{r} \]

4. **Convert relative error to percentage**:
To find the percentage error in the area, we multiply the relative error by 100:
\[ \text{Percentage error in area} = \frac{dA}{A} \times 100 = \frac{2dr}{r} \times 100 \]

5. **Substitute the given error in radius**:
We are given that the error in the radius \( dr \) is 3%. Therefore, we can substitute:
\[ \text{Percentage error in area} = 2 \times 3\% \]
This results in:
\[ \text{Percentage error in area} = 6\% \]

### Final Answer:
The error in the measurement of the area of the circle is **6%**.

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