The length and breadth of a rectangular sheet are 16.2 cm and 10.1cm, respectively. The area of the sheet in appropriate significant figures and error is

Options

A. `164+-3cm^(2)`
B. `163.62+-2.6cm^(2)`
C. `163.6+-2.6cm^(2)`
D. `163.62+-3cm^(2)`

Explanation

To find the area of a rectangular sheet given its length and breadth, we will follow these steps:

### Step 1: Identify the given values
- Length (L) = 16.2 cm
- Breadth (B) = 10.1 cm

### Step 2: Calculate the area
The area (A) of a rectangle is calculated using the formula:
\[ A = L \times B \]

Substituting the values:
\[ A = 16.2 \, \text{cm} \times 10.1 \, \text{cm} \]
\[ A = 163.62 \, \text{cm}^2 \]

### Step 3: Determine the significant figures
- The length (16.2 cm) has 3 significant figures.
- The breadth (10.1 cm) also has 3 significant figures.

When multiplying, the result should be expressed in the least number of significant figures, which in this case is 3.

### Step 4: Round the area to the appropriate significant figures
Rounding 163.62 cm² to 3 significant figures gives:
\[ A \approx 164 \, \text{cm}^2 \]

### Step 5: Calculate the error in the area
To find the error in the area, we will use the formula for error propagation in multiplication:
\[ dA = A \left( \frac{dL}{L} + \frac{dB}{B} \right) \]
Where:
- \( dA \) = error in area
- \( dL \) = error in length
- \( dB \) = error in breadth

Assuming the least count (error) for both length and breadth is 0.1 cm:
- \( dL = 0.1 \, \text{cm} \)
- \( dB = 0.1 \, \text{cm} \)

Now substituting the values:
\[ dA = 163.62 \, \text{cm}^2 \left( \frac{0.1}{16.2} + \frac{0.1}{10.1} \right) \]

Calculating the fractions:
\[ \frac{0.1}{16.2} \approx 0.00617 \]
\[ \frac{0.1}{10.1} \approx 0.00990 \]

Adding these:
\[ 0.00617 + 0.00990 \approx 0.01607 \]

Now, substituting back:
\[ dA \approx 163.62 \, \text{cm}^2 \times 0.01607 \approx 2.63 \, \text{cm}^2 \]

### Step 6: Round the error to appropriate significant figures
Since the error should be expressed in one significant figure (as per convention), we round 2.63 cm² to:
\[ dA \approx 3 \, \text{cm}^2 \]

### Final Result
Thus, the area of the rectangular sheet with its error is:
\[ A = 164 \pm 3 \, \text{cm}^2 \]