The length and breadth of a field are 22.4 cm and 15.8 cm respectively and have been measured to an accuracy of 0.2 em. Find the percentage error in the area of the field.

The correct option is 2.16%.

Step-by-step Explanation:

We are given the following measurements for the field:
Length of the field, $L=22.4\text{cm}$
Breadth of the field, $B=15.8\text{cm}$
The accuracy of the measurement (which represents the absolute error in both length and breadth) is:
$\Delta L=\Delta B=0.2\text{cm}$

The area ($A$) of a rectangular field is calculated as:
$A=L\times B$

For a product of two quantities, the relative error in the calculated area is the sum of the relative errors of the individual measured dimensions. Therefore, the fractional error is:
$\frac{\Delta A}{A}=\frac{\Delta L}{L}+\frac{\Delta B}{B}$

To find the percentage error in the area, we multiply the fractional error by 100:
$\text{Percentage Error in}A=\left(\frac{\Delta L}{L}+\frac{\Delta B}{B}\right)\times 100$

Substituting the given values into the equation:
$\text{Percentage Error}=\left(\frac{0.2}{22.4}+\frac{0.2}{15.8}\right)\times 100$

Calculating the fractions:
$\frac{0.2}{22.4}\approx 0.00893$
$\frac{0.2}{15.8}\approx 0.01266$

Adding these individual fractional errors together:
$0.00893+0.01266=0.02159$

Multiplying by 100 to convert to a percentage:
$0.02159\times 100=2.159\%\approx 2.16\%$

Thus, the percentage error in the area of the field is approximately 2.16%.