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 when the circumference is measured with an error, we can follow these steps:

### Step 1: Understand the given data
- The circumference \( P \) of the circle is given as \( 56 \) cm.
- The error in circumference \( dP \) is \( 0.02 \) cm.

### Step 2: Use the formula for circumference
The formula for the circumference of a circle is:
\[ P = 2\pi r \]
where \( r \) is the radius of the circle.

### Step 3: Differentiate the circumference formula
Differentiating both sides with respect to \( r \):
\[ dP = 2\pi dr \]
where \( dr \) is the error in the radius.

### Step 4: Relate the errors
From the differentiation, we can express the relative error in circumference as:
\[ \frac{dP}{P} = \frac{2\pi dr}{2\pi r} = \frac{dr}{r} \]

### Step 5: Calculate the relative error in circumference
Substituting the known values:
\[ \frac{dP}{P} = \frac{0.02}{56} \]
Calculating this gives:
\[ \frac{dP}{P} = \frac{0.02}{56} = \frac{1}{2800} \]

### Step 6: Relate the error in area to the error in radius
The area \( A \) of the circle is given by:
\[ A = \pi r^2 \]
Differentiating this with respect to \( r \):
\[ dA = 2\pi r \, dr \]
Now, the relative error in area can be expressed as:
\[ \frac{dA}{A} = \frac{2\pi r \, dr}{\pi r^2} = \frac{2dr}{r} \]

### Step 7: Substitute the relative error in radius
From Step 4, we found that \( \frac{dr}{r} = \frac{1}{2800} \). Therefore:
\[ \frac{dA}{A} = 2 \cdot \frac{1}{2800} = \frac{2}{2800} = \frac{1}{1400} \]

### Step 8: Calculate the percentage error in area
To find the percentage error in area:
\[ \text{Percentage error in area} = \frac{dA}{A} \times 100 = \frac{1}{1400} \times 100 = \frac{100}{1400} = \frac{1}{14} \% \]

### Final Answer
The percentage error in the area of the circle is \( \frac{1}{14} \% \).

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