Physics

A physical quantity $$x$$ is calculated from the relation $$x={ a }^{ 3 }{ b }^{ 2 }/\sqrt { cd }$$. Calculate percentage error in $$x$$, if $$a,b,c$$ and $$d$$ are measured respectively with an error of $$1\%$$, $$3\%$$, $$4\%$$ and $$2\%$$.


ANSWER

$$12\%$$.


SOLUTION
Given $$ x = \dfrac{a^3b^2}{c^{1/2}d^{1/2}}$$
On taking log and differentiating, we get

$$\dfrac{\Delta x}{x}$$ $$=3\dfrac{\Delta a}{a}$$  $$+2\dfrac{\Delta b}{b}$$  $$+(\dfrac{1}{2})\dfrac{\Delta c}{c}$$  $$+(\dfrac{1}{2})\dfrac{\Delta d}{d}$$ 

$$\Rightarrow $$ Percentage error in $$ x= 3(1)+2(3)+\dfrac{1}{2}(4)+\dfrac{1}{2}(2) = 12\%$$
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