Question

The sides of a rectangle are \[\left( {10.5 \pm 0.2} \right)cm\]and \[\left( {5.2 \pm 0.1} \right)cm\]. Calculate its perimeter with its error limits.

Answer 1

Hint: The question is related to error limits, the errors are solved separately from the required value such as perimeter or area etc. The formula we will use for the perimeter of the rectangle is \[P = 2\left( {l + b} \right)\] where \[l\] is the length of the rectangle and \[b\] is the breadth of the rectangle. Also, the same formula is used by putting the values of error limits respectively and then adding and subtracting them to get the resultant perimeter.

Complete step-by-step solution:
Errors are something that are unavoidable as when it’s a matter of measuring different quantities so, due to the error in measuring instruments or because of any other result, error limits are used.
Given : \[l = 10.5cm\] \[\Delta l = 0.2cm\], \[b = 5.2cm\]\[\Delta b = 0.1cm\]
Here \[\Delta \]represents the absolute error.
Now for finding the perimeter of the rectangle we use formula
 \[P = 2\left( {l + b} \right)\]
Now putting the values we get,
\[P = 2\left( {10.5 + 5.2} \right)cm\]
On solving we get,
\[P = 2\left( {15.7} \right)cm\]
On simplifying we get,
\[P = 31.4cm\]
Now solving the errors limits by same formula we get,
\[\Delta P = 2\left( {\Delta l + \Delta b} \right)\]
Now putting the values we get,
\[\Delta P = 2\left( {0.2 + 0.1} \right)cm\]
On solving we get,
\[\Delta P = 2\left( {0.3} \right)cm\]
On simplifying we get,
\[\Delta P = 0.6cm\]
The resultant perimeter with error limits will be given by :
\[ = P \pm \Delta P\]
On putting the values we get,
\[ = \left( {31.4 \pm 0.6} \right)cm\]

Note: After finding the perimeter with error limits do not forget to write the unit in which the dimensions are given. While addition or subtraction with error absolute errors are always added to each other. For example, say \[\left( {A \pm \Delta A} \right) + \left( {B \pm \Delta B} \right) = \left[ {\left( {A + B} \right) \pm \left( {\Delta A + \Delta B} \right)} \right]\] and \[\left( {A \pm \Delta A} \right) - \left( {B \pm \Delta B} \right) = \left[ {\left( {A - B} \right) \pm \left( {\Delta A + \Delta B} \right)} \right]\]. So, in both the cases the absolute error is added with each other irrespective of addition or subtraction.