Question

A physical quantity $\rho$ is related to four variables $\alpha ,\beta ,\gamma$ and $\eta$ as $\rho =\frac{{\alpha }^3{\beta }^2}{\sqrt{\gamma \eta }}$ The percentage errors of measurements in $\alpha ,\beta ,\gamma$ and $\eta$ are 1%, 3%, 4% and 2% respectively. Find the percentage error in $\rho$.

• A. 6.5%
• B. 12%
• C. 21%
• D. 26%

Answer 1

The correct option is B 12%

The physical quantity is given as, $P=\frac{{\alpha }^2{\beta }^2}{\sqrt{\gamma \eta }}$

Taking log on both sides and differentiating, $\therefore \frac{\Delta P}{P}\times 100=3\frac{\Delta \alpha }{\alpha }\times 100+2\frac{\Delta \beta }{\beta }\times 100+\frac{1}{2}\frac{\Delta \gamma }{\gamma }\times 100+\frac{1}{2}\frac{\Delta \eta }{\eta }\times 100$

$⟹\frac{\Delta P}{P}\times 100=3\times 1+2\times 3+\frac{1}{2}\times 4+\frac{1}{2}\times 2=12$%

Thus percentage error in $P$ is equal to $12$%.