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

Explanation

To solve the problem of finding the area of a rectangular sheet with given dimensions and their associated errors, we can follow these steps:

### Step 1: Identify the given dimensions and their uncertainties
- Length (L) = 16.2 cm with an uncertainty (ΔL) of ±0.1 cm
- Breadth (B) = 10.1 cm with an uncertainty (ΔB) of ±0.1 cm

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

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

### Step 3: Determine the significant figures
The numbers 16.2 and 10.1 both have three significant figures. Therefore, the area should also be reported with three significant figures:
\[ A \approx 164 \, \text{cm}^2 \]

### Step 4: Calculate the relative error in area
The relative error in area can be calculated using the formula:
\[ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} \]

Substituting the values:
- ΔL = 0.1 cm
- L = 16.2 cm
- ΔB = 0.1 cm
- B = 10.1 cm

Calculating each term:
\[ \frac{\Delta L}{L} = \frac{0.1}{16.2} \approx 0.00617 \]
\[ \frac{\Delta B}{B} = \frac{0.1}{10.1} \approx 0.00990 \]

Adding these relative errors:
\[ \frac{\Delta A}{A} \approx 0.00617 + 0.00990 \approx 0.01607 \]

### Step 5: Calculate the absolute error in area
Now, we can find the absolute error (ΔA) in the area:
\[ \Delta A = A \times \frac{\Delta A}{A} \]
\[ \Delta A \approx 163.62 \, \text{cm}^2 \times 0.01607 \approx 2.63 \, \text{cm}^2 \]

### Step 6: Round the error to appropriate significant figures
Since the area is reported with three significant figures, we round the error to one significant figure:
\[ \Delta A \approx 3 \, \text{cm}^2 \]

### Step 7: Final result
The area of the rectangular sheet with its uncertainty is:
\[ A = 164 \pm 3 \, \text{cm}^2 \]

### Summary of the solution
The area of the rectangular sheet is \( 164 \pm 3 \, \text{cm}^2 \).

---