If an error of `1^(@)` is made in measuring the angle of of a sector of radius 30 cm, then the approximate eror in its area, is

Options

A. `450 cm^(2)`
B. `25 pi cm^(2)`
C. `2.5 pi cm^(2)`
D. none of these

Explanation

To solve the problem of finding the approximate error in the area of a sector when an error of \(1^\circ\) is made in measuring the angle, we can follow these steps:

### Step 1: Write the formula for the area of a sector
The area \(A\) of a sector with radius \(r\) and angle \(\theta\) (in degrees) is given by:
\[ A = \frac{\pi r^2 \theta}{360} \]

### Step 2: Differentiate the area with respect to the angle
To find the relationship between the error in the area and the error in the angle, we differentiate \(A\) with respect to \(\theta\):
\[ \frac{dA}{d\theta} = \frac{\pi r^2}{360} \]

### Step 3: Substitute the radius value
Given that the radius \(r = 30\) cm, we substitute this value into the derivative:
\[ \frac{dA}{d\theta} = \frac{\pi (30)^2}{360} = \frac{\pi \cdot 900}{360} = \frac{25\pi}{10} = 25\pi \text{ cm}^2/\text{degree} \]

### Step 4: Convert the angle error from degrees to radians
The error in measuring the angle is given as \(1^\circ\). To use this in our calculations, we need to convert degrees to radians:
\[ \Delta \theta = 1^\circ = \frac{\pi}{180} \text{ radians} \]

### Step 5: Calculate the approximate error in area
Now we can find the approximate error in the area \(\Delta A\) using the formula:
\[ \Delta A \approx \frac{dA}{d\theta} \cdot \Delta \theta \]
Substituting the values we found:
\[ \Delta A \approx 25\pi \cdot \frac{\pi}{180} \]
\[ \Delta A \approx \frac{25\pi^2}{180} \text{ cm}^2 \]

### Step 6: Simplify the expression
To simplify \(\frac{25\pi^2}{180}\):
\[ \Delta A \approx \frac{5\pi^2}{36} \text{ cm}^2 \]

### Final Answer
The approximate error in the area of the sector is:
\[ \Delta A \approx \frac{5\pi^2}{36} \text{ cm}^2 \]