The radius of a circle is measured to be `(10.8+-0.3)m`. Calcualte the area of the circle.

Explanation

To solve the problem of calculating the area of a circle given the radius and its uncertainty, we can follow these steps:

### Step 1: Understand the formula for the area of a circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Substitute the value of the radius
Given the radius \( r = 10.8 \, \text{m} \), we can substitute this value into the area formula. We will use \( \pi \approx \frac{22}{7} \) for our calculations:
\[ A = \frac{22}{7} \times (10.8)^2 \]

### Step 3: Calculate \( (10.8)^2 \)
First, calculate the square of the radius:
\[ (10.8)^2 = 116.64 \]

### Step 4: Calculate the area
Now substitute \( 116.64 \) back into the area formula:
\[ A = \frac{22}{7} \times 116.64 \]

Calculating this gives:
\[ A \approx 22 \times 16.663 = 366.586 \, \text{m}^2 \]

### Step 5: Round off the area
Rounding \( 366.586 \) to one decimal place gives:
\[ A \approx 366.6 \, \text{m}^2 \]

### Step 6: Calculate the uncertainty in the area
To find the uncertainty in the area \( \Delta A \), we use the formula for the propagation of uncertainty:
\[ \frac{\Delta A}{A} = 2 \frac{\Delta r}{r} \]
where \( \Delta r = 0.3 \, \text{m} \) and \( r = 10.8 \, \text{m} \).

### Step 7: Substitute the values for uncertainty
Substituting the values:
\[ \frac{\Delta A}{366.6} = 2 \times \frac{0.3}{10.8} \]

Calculating the right side:
\[ \frac{\Delta A}{366.6} = 2 \times 0.02778 \approx 0.055 \]

### Step 8: Calculate \( \Delta A \)
Now, calculate \( \Delta A \):
\[ \Delta A = 0.055 \times 366.6 \approx 20.4 \, \text{m}^2 \]

### Step 9: Final result including uncertainty
The final area of the circle with its uncertainty is:
\[ A = 366.6 \pm 20.4 \, \text{m}^2 \]

### Summary of the Solution
The area of the circle is approximately \( 366.6 \, \text{m}^2 \) with an uncertainty of \( \pm 20.4 \, \text{m}^2 \).

---