Question

A physical parameter $a$ can be determined by measuring the parameters $b, c, d$ and $e$ using the relation $a=b^{\alpha} c^{\beta} / d^{\gamma} e^{\delta}$. If the maximum errors in the measurement of $b, c, d$ and $e$ are $b_{1} \%, c_{1} \%, d_{1} \%$ and $e^{1} \%$, then the maximum error in the value of $a$ determined by the experiment is

• A. $\left(b_{1}+c_{1}+d_{1}+e_{1}\right) \%$
• B. $\left(b_{1}+c_{1}-d_{1}-e_{1}\right) \%$
• C. $\left(\alpha b_{1}+\beta c_{1}-\gamma d_{1}-\delta e_{1}\right) \%$
• D. $\left(a b_{1}+\beta c_{1}+\gamma d_{1}+\delta e_{1}\right) \%$

Answer 1

Answer: (D)

$a=b^{\alpha} c^{\beta} / d^{\gamma} e^{\delta}$
So, maximum error in $a$ is given by
$\left(\frac{\Delta a}{a} \times 100\right)_{\max }=\alpha \cdot \frac{\Delta b}{b} \times 100+\beta \cdot \frac{\Delta c}{c} \times 100$
$+\gamma \cdot \frac{\Delta d}{d} \times 100+\delta \cdot \frac{\Delta e}{e} \times 100$
$=\left(\alpha b_{1}+\beta c_{1}+\gamma d_{1}+\delta e_{1}\right) \%$