# A physical quantity z depends on four observables a, b, c

Question:

A physical quantity $z$ depends on four observables $a, b, c$

and $d$, as $z=\frac{a^{2} b^{\frac{2}{3}}}{\sqrt{c} d^{3}} .$ The percentages of error in the

measurement of $a, b, c$ and $d$ are $2 \%, 1.5 \%, 4 \%$ and $2.5 \%$ respectively. The percentage of error in $z$ is :

1. (1) $12.25 \%$

2. (2) $16.5 \%$

3. (3) $13.5 \%$

4. (4) $14.5 \%$

Correct Option: , 4

Solution:

(4) Given : $Z=\frac{a^{2} b^{2 / 3}}{\sqrt{c} d^{3}}$

Percentage error in $Z$,

$=\frac{\Delta Z}{Z}=\frac{2 \Delta a}{a}+\frac{2}{3} \frac{\Delta b}{b}+\frac{1}{2} \frac{\Delta c}{c}+\frac{3 \Delta d}{d}$

$=2 \times 2+\frac{2}{3} \times 1.5+\frac{1}{2} \times 4+3 \times 2.5=14.5 \%$