Question

A physical quantity, $y=\frac{a^4{b}^2}{(c{d}^4{)}^{1/3}}$ has four observables a, b, c and d. The percentage error in a, b, c and d are 2%, 3%, 4% and 5% respectively the error in y will be

• A. 6%
• B. 11%
• C. 12%
• D. 22%

Answer 1

The correct option is D 22%

Here $y=\frac{a^4{b}^2}{(c{d}^4{)}^{1/3}}$
Taking log on both sides, we get
$logy=4loga+2logb-\frac{1}{3}logc-\frac{4}{3}logd$
Differentiating both sides to calculate error,
$\frac{\Delta y}{y}\times 100=4(\frac{\Delta a}{a}\times 100)+2(\frac{\Delta d}{d}\times 100)+\frac{1}{3}(\frac{\Delta a}{a}\times 100)+\frac{4}{3}(\frac{\Delta d}{d}\times 100)=[4\times 2\%+2\times 3\%+\frac{1}{3}\times 4\%+\frac{4}{3}\times 5\%]=22\%$