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. `164pm3" cm"^(2)`
B. `163.62pm2.6" cm"^(2)`
C. `163.6pm2.6" cm"^(2)`
D. `163.62pm3" cm"^(2)`

Explanation

To solve the problem of finding the area of a rectangular sheet given its length and breadth, and to determine the area with the appropriate significant figures and error, we can follow these steps:

### Step 1: Calculate the Area
The area \( A \) of a rectangle is calculated using the formula:
\[ A = \text{length} \times \text{breadth} \]
Given:
- Length \( L = 16.2 \, \text{cm} \)
- Breadth \( B = 10.1 \, \text{cm} \)

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

### Step 2: Round the Area to Appropriate Significant Figures
Next, we need to round the area to the appropriate number of significant figures. The length (16.2 cm) has 3 significant figures, and the breadth (10.1 cm) also has 3 significant figures. Therefore, the area should be rounded to 3 significant figures:
\[ A \approx 164 \, \text{cm}^2 \]

### Step 3: Calculate the Absolute Error in Area
To find the error in the area, we first need to determine the absolute errors in length and breadth. The least count (or the smallest division) of the measuring instrument used for length and breadth is assumed to be 0.1 cm.

Thus:
- Absolute error in length \( \Delta L = 0.1 \, \text{cm} \)
- Absolute error in breadth \( \Delta B = 0.1 \, \text{cm} \)

The formula for the relative error in area is given by:
\[ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} \]

Substituting the values:
\[ \frac{\Delta A}{A} = \frac{0.1}{16.2} + \frac{0.1}{10.1} \]

Calculating each term:
\[ \frac{0.1}{16.2} \approx 0.00617 \quad \text{and} \quad \frac{0.1}{10.1} \approx 0.00990 \]
Adding these together:
\[ \frac{\Delta A}{A} \approx 0.00617 + 0.00990 \approx 0.01607 \]

### Step 4: Calculate the Absolute Error in Area
Now, we can calculate the absolute error \( \Delta A \):
\[ \Delta A = A \times \frac{\Delta A}{A} = 164 \, \text{cm}^2 \times 0.01607 \approx 2.63 \, \text{cm}^2 \]

### Step 5: Round the Error to Appropriate Significant Figures
Since the error should be rounded to one significant figure:
\[ \Delta A \approx 3 \, \text{cm}^2 \]

### Step 6: Present the Final Result
Finally, we can express the area with its error:
\[ A = 164 \pm 3 \, \text{cm}^2 \]

### Summary of the Solution
The area of the rectangular sheet, with appropriate significant figures and error, is:
\[ \boxed{164 \pm 3 \, \text{cm}^2} \]