The length and breadth of a rectangle are `( 5.7 +- 0.1 ) cm and ( 3.4 +- 0.2 ) cm`, respectively calculate the area of rectangle with error limits.

Explanation

To calculate the area of a rectangle with given dimensions and their associated uncertainties, we can follow these steps:

### Step 1: Write down the dimensions and their uncertainties
- Length (L) = \(5.7 \, \text{cm} \pm 0.1 \, \text{cm}\)
- Breadth (B) = \(3.4 \, \text{cm} \pm 0.2 \, \text{cm}\)

### Step 2: Calculate the area (A) of the rectangle
The area of a rectangle is given by the formula:
\[ A = L \times B \]
Substituting the values:
\[ A = 5.7 \, \text{cm} \times 3.4 \, \text{cm} = 19.38 \, \text{cm}^2 \]

### Step 3: Calculate the relative error in area
The formula for the relative error in multiplication is:
\[ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} \]
Where:
- \(\Delta L\) = uncertainty in length = \(0.1 \, \text{cm}\)
- \(\Delta B\) = uncertainty in breadth = \(0.2 \, \text{cm}\)

Calculating the relative errors:
\[ \frac{\Delta L}{L} = \frac{0.1}{5.7} \approx 0.01754 \]
\[ \frac{\Delta B}{B} = \frac{0.2}{3.4} \approx 0.05882 \]

Now, adding these relative errors:
\[ \frac{\Delta A}{A} = 0.01754 + 0.05882 \approx 0.07636 \]

### Step 4: Calculate the absolute error in area
Now, we can find the absolute error in the area using:
\[ \Delta A = A \times \frac{\Delta A}{A} \]
Substituting the values:
\[ \Delta A = 19.38 \, \text{cm}^2 \times 0.07636 \approx 1.48 \, \text{cm}^2 \]

### Step 5: Round the error to one significant figure
Since the uncertainty \(1.48\) can be rounded to \(1.5\) for significant figures.

### Step 6: Write the final result
Thus, the area of the rectangle with its error limits is:
\[ A = 19.38 \pm 1.5 \, \text{cm}^2 \]

### Final Answer
The area of the rectangle is \(19.4 \pm 1.5 \, \text{cm}^2\) (rounded to one decimal place).

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