The circumference of a circle is measured as 56 cm with an error 0.02 cm. The percentage error in its area is

Options

A. `1//7`
B. `1//28`
C. `1//14`
D. `1//56`

Explanation

To find the percentage error in the area of a circle given the circumference and its error, we can follow these steps:

### Step 1: Understand the relationship between circumference and radius
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
Given that the circumference is measured as \( 56 \, \text{cm} \), we can set up the equation:
\[ 2\pi r = 56 \]

### Step 2: Solve for the radius \( r \)
Rearranging the equation to find \( r \):
\[ r = \frac{56}{2\pi} = \frac{28}{\pi} \, \text{cm} \]

### Step 3: Determine the error in the radius
The error in the circumference \( \Delta C \) is given as \( 0.02 \, \text{cm} \). The relationship between the error in circumference and the error in radius is given by:
\[ \Delta C = 2\pi \Delta r \]
Substituting the known error:
\[ 0.02 = 2\pi \Delta r \]
Solving for \( \Delta r \):
\[ \Delta r = \frac{0.02}{2\pi} = \frac{0.01}{\pi} \, \text{cm} \]

### Step 4: Calculate the area of the circle
The area \( A \) of the circle is given by:
\[ A = \pi r^2 \]
Substituting \( r = \frac{28}{\pi} \):
\[ A = \pi \left(\frac{28}{\pi}\right)^2 = \pi \cdot \frac{784}{\pi^2} = \frac{784}{\pi} \, \text{cm}^2 \]

### Step 5: Determine the change in area \( \Delta A \)
Using the formula for the change in area:
\[ \Delta A = 2\pi r \Delta r \]
Substituting \( r = \frac{28}{\pi} \) and \( \Delta r = \frac{0.01}{\pi} \):
\[ \Delta A = 2\pi \left(\frac{28}{\pi}\right) \left(\frac{0.01}{\pi}\right) \]
Simplifying:
\[ \Delta A = 2 \cdot 28 \cdot 0.01 \cdot \frac{1}{\pi} = \frac{56 \cdot 0.01}{\pi} = \frac{0.56}{\pi} \, \text{cm}^2 \]

### Step 6: Calculate the percentage error in the area
The percentage error in the area is given by:
\[ \text{Percentage error} = \left(\frac{\Delta A}{A}\right) \times 100 \]
Substituting \( \Delta A = \frac{0.56}{\pi} \) and \( A = \frac{784}{\pi} \):
\[ \text{Percentage error} = \left(\frac{\frac{0.56}{\pi}}{\frac{784}{\pi}}\right) \times 100 = \left(\frac{0.56}{784}\right) \times 100 \]
Calculating:
\[ \text{Percentage error} = \frac{0.56 \times 100}{784} = \frac{56}{784} = \frac{1}{14} \approx 0.0714 \times 100 \approx 0.714\% \]

Thus, the percentage error in the area is:
\[ \text{Percentage error} \approx 0.714\% \]