Question

A physical quantity $Q$ is released to four observable $x,y,z$ and $t$ by the relation
$Q=\frac{x^{2/5}{z}^3}{y\sqrt{t}}$
The percentage errors of measurement in $x,y,z$ and $t$ are $2.5\%,2\%,0.5\%$ and $1\%$ receptively. The percentage error in $Q$ will be

• A. $5\%$
• B. $4.5\%$
• C. $8\%$
• D. $7.75\%$

Answer 1

The correct option is A $5\%$

$Q=\frac{x^{2/5}{z}^3}{y\sqrt{t}}$

$\begin{array}{cc}\text{Given:}& \frac{\delta x}{x}\times 100\%=2.5\%\\ & \frac{\delta z}{z}\times 100\%=0.5\%\end{array}$

$\begin{array}{c}\frac{\delta y}{y}\times 100\%=2\%\\ \frac{\delta t}{t}\times 100\%=1\%\end{array}$

$\begin{array}{c}\text{Taking Log both side in}e{q}^n\left(i\right)\\ \log Q=\log \left(\frac{x^{2/5}{z}^3}{y{t}^{\prime \prime 2}}\right)\\ \log Q=\frac{2}{5}\log x+3\log z-\log y-\frac{1}{2}\log t\end{array}$

$\begin{array}{c}\text{Differentiating both side}\\ \frac{\delta Q}{Q}=\frac{2}{5}\frac{\delta x}{x}+3\frac{\delta z}{z}-\frac{\delta y}{y}-\frac{1}{2}\frac{\delta t}{t}\end{array}$

$\text{Error is always added}$

$\frac{\delta Q_1}{Q}\times 100\%=\frac{2}{5}\frac{\delta x}{x}\times {100}^{\%}+3\frac{\delta z}{2}\times 100\%+\frac{\delta y\times 100\%}{y}+\frac{1}{2}\frac{\delta t\times 100}{t}$$=5\%$