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

The correct option is 164 ± 3 cm².

Step 1: Identify the given values and their associated uncertainties (least counts).
The measured length ($l$) and breadth ($b$) of the rectangular sheet are given as:
$l=16.2\text{cm}$
$b=10.1\text{cm}$
Since both measurements are recorded to one decimal place, the least count (absolute error) in both measurements is:
$\Delta l=0.1\text{cm}$
$\Delta b=0.1\text{cm}$

Step 2: Calculate the area and apply the rules of significant figures.
The area ($A$) of the rectangular sheet is given by:
$A=l\times b$
Substituting the values:
$A=16.2\text{cm}\times 10.1\text{cm}=163.62{\text{cm}}^2$
In multiplication, the final result should be rounded off to contain the same number of significant figures as the measurement with the least number of significant figures. Here, both $l$ (16.2) and $b$ (10.1) have 3 significant figures. Therefore, the area must be rounded off to 3 significant figures:
$A\approx 164{\text{cm}}^2$

Step 3: Calculate the propagated error in the area.
For multiplication, the relative error in the area is the sum of the relative errors in the individual dimensions:
$\frac{\Delta A}{A}=\frac{\Delta l}{l}+\frac{\Delta b}{b}$
Rearranging the formula to find the absolute error ($\Delta A$) using the unrounded value of area:
$\Delta A=A\times (\frac{\Delta l}{l}+\frac{\Delta b}{b})$
$\Delta A=163.62\times (\frac{0.1}{16.2}+\frac{0.1}{10.1})$
$\Delta A=163.62\times (0.00617+0.00990)$
$\Delta A=163.62\times 0.01607\approx 2.63{\text{cm}}^2$

Step 4: Round off the error to match the precision of the measured value.
Since the value of the area ($164{\text{cm}}^2$) is rounded to the nearest unit (integer) place, the absolute error must also be rounded to the same decimal place (units place).
Rounding $2.63{\text{cm}}^2$ to the nearest integer gives:
$\Delta A\approx 3{\text{cm}}^2$

Therefore, the area of the sheet in appropriate significant figures and error is:
$A\pm \Delta A=164\pm 3{\text{cm}}^2$