Question

The length and breadth of the rectangle are \[\left( 5.7\pm 0.1 \right)\] cm and \[\left( 3.4\pm 0.2 \right)\] cm. Calculate the area of the rectangle with error limits.
A. \[\left( 19.0\pm 5.5 \right)\]
B. \[\left( 39.0\pm 1.5 \right)\]
C. \[\left( 29.0\pm 2.5 \right)\]
D. \[\left( 19.0\pm 1.5 \right)\]

Answer 1

Hint: To solve this question we must know the formula to find error limits. And it is assumed that students know the formula to find the area of a rectangle. So after finding the area of the rectangle and the error limits we must add them to obtain the required result. While propagation of errors in addition as well as subtraction the absolute error says $\Delta$ x and $\Delta$y will always be added.

Formula used: \[A=l\times b\]
 \[\dfrac{\Delta A}{A}=\pm \left[ \dfrac{\Delta l}{l}+\dfrac{\Delta b}{b} \right]\]

Complete step by step answer:
Error is something that should not be done. But, there is no way to avoid certain errors. Errors can happen during calculation, due to the error in measuring instruments or because of any other result. So, error limits are used. Error limits are basically the maximum and minimum deviation from the result that should be obtained.
We know that,
\[A=l\times b\]
\[A=5.7\times 3.4=19.38\]cm
Now, using the formula for relative error to calculate the absolute error \[\Delta A\]
We get,
\[\dfrac{\Delta A}{A}=\pm \left[ \dfrac{\Delta l}{l}+\dfrac{\Delta b}{b} \right]\]
Substituting the given values in above equation
We get,
\[\dfrac{\Delta A}{19.38}=\pm \left[ \dfrac{0.1}{5.7}+\dfrac{0.2}{3.4} \right]\]
Multiplying by 19.38 to both the sides
 We get,
\[\Delta A=\pm \left[ \dfrac{1.48}{19.38}\times 19.38 \right]\]
On solving above equation
We get,
\[\Delta A=\pm 1.48\]
Now, rounding up to two significant figures.
We get,
\[\Delta A=\pm 1.5\]
Therefore the area of rectangle with error limits will be given by the formula,
\[A+\Delta A\]
We have rounded up the error limit up to two significant figures
Therefore we must round up the error up to two significant digits as well
which gives us,
A = 19.0
Now substituting the area and error limit in above formula
We get,
\[19.0\pm 1.5\]

So, the correct answer is “Option D”.

Additional Information: Relative error is the ratio of absolute error to mean value of the given quantity. Similarly, absolute error is given by the difference between individual measurement and true value of the quantity.

Note: Students must know the concept of significant figures. They must be careful in rounding up the numbers and apply the rules for significant figures. While addition or subtraction with error absolute errors are always added to each other. For example, say \[(A\pm \Delta A)+(B+\Delta B)\]\[=[(A+B)\pm (\Delta A+\Delta B)]\] and \[(A\pm \Delta A)-(B+\Delta B)\]\[=[(A-B)\pm (\Delta A+\Delta B)]\]. So in both these cases absolute error is added irrespective of addition or subtraction.