The lengths and breadth of a rectangle are `(5.7 +- 0.1)cm and (2.4 +- 0.2)` cm. Calculate area of the rectangle with error limits.

Explanation

To calculate the area of a rectangle with given lengths and breadth along with their uncertainties, we can follow these steps:

### Step 1: Identify the given values
The length \( L \) and breadth \( B \) of the rectangle are given as:
- Length \( L = 5.7 \pm 0.1 \) cm
- Breadth \( B = 2.4 \pm 0.2 \) cm

### Step 2: Calculate the area \( A \)
The area \( A \) of the rectangle is calculated using the formula:
\[ A = L \times B \]
Substituting the values:
\[ A = 5.7 \, \text{cm} \times 2.4 \, \text{cm} = 13.68 \, \text{cm}^2 \]
Rounding off to one decimal place (since the least precise measurement has one decimal place):
\[ A \approx 13.7 \, \text{cm}^2 \]

### Step 3: Calculate the relative uncertainties
To find the uncertainty in the area \( \Delta A \), we use the formula for the propagation of uncertainties for multiplication:
\[ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} \]
Where:
- \( \Delta L = 0.1 \, \text{cm} \)
- \( \Delta B = 0.2 \, \text{cm} \)

Calculating the relative uncertainties:
\[ \frac{\Delta L}{L} = \frac{0.1}{5.7} \approx 0.01754 \]
\[ \frac{\Delta B}{B} = \frac{0.2}{2.4} \approx 0.08333 \]

### Step 4: Combine the relative uncertainties
Now, add the relative uncertainties:
\[ \frac{\Delta A}{A} = 0.01754 + 0.08333 \approx 0.10087 \]

### Step 5: Calculate the absolute uncertainty \( \Delta A \)
Now, multiply the relative uncertainty by the area to find the absolute uncertainty:
\[ \Delta A = A \times \frac{\Delta A}{A} = 13.7 \, \text{cm}^2 \times 0.10087 \approx 1.38 \, \text{cm}^2 \]
Rounding off to one decimal place gives:
\[ \Delta A \approx 1.4 \, \text{cm}^2 \]

### Step 6: Final result
The area of the rectangle with its error limits is:
\[ A = 13.7 \pm 1.4 \, \text{cm}^2 \]

### Summary
The area of the rectangle is \( 13.7 \, \text{cm}^2 \) with an uncertainty of \( \pm 1.4 \, \text{cm}^2 \).

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