question_answer 69)

                  A physical quantity \[X\] is related to four measureable quantities a, b, c and d as follows:                 \[X={{a}^{2}}{{b}^{3}}{{c}^{{5}/{2}\;}}{{d}^{-2}}\]                 The percentage error in the measurement of a, b, c and d are 1%, 2% and 4%, respectively. What is the percentage error in quantity \[X\]? If the value of \[X\] calculated on the basis of the above relation is 2.763, to what value should you round off the result.                

Answer:

                  \[\frac{\Delta X}{X}=2\frac{\Delta a}{a}+3\frac{\Delta b}{b}+\frac{5}{2}\frac{\Delta c}{c}+2\frac{\Delta d}{d}\]                 \[=2\left( \frac{1}{100} \right)+3\left( \frac{2}{100} \right)+\frac{5}{2}\left( \frac{3}{100} \right)+2\left( \frac{4}{100} \right)\]                 \[=0.235=0.24\]                 Percentage error in                 \[X=\left( \frac{\Delta X}{X} \right)100=(0.24)(100)\]                 \[=24%\]                 Since the error (0.24) is in the first decimal place, the result (2.763) should be rounded off to 2.8.