Question

A physical quantity P is related to four observables a, b, c and d as follows:
P = a^3b^3/underroot cd

The percentage errors of the measurement in a, b, c and d are 1%, 3%, 4% and 2% respectively. What is the percentage error in the quantity P? ​

Answer 1

Answer

Answer= 3x1 +2x3 + ½ x 4 + 2

Answer= 3x1 +2x3 + ½ x 4 + 2= 3 + 6 +2 + 2

Answer= 3x1 +2x3 + ½ x 4 + 2= 3 + 6 +2 + 2= 13 %

Answer= 3x1 +2x3 + ½ x 4 + 2= 3 + 6 +2 + 2= 13 %Percentage error in P = 13 %

Answer= 3x1 +2x3 + ½ x 4 + 2= 3 + 6 +2 + 2= 13 %Percentage error in P = 13 %Value of P is given as 3.763.

Answer= 3x1 +2x3 + ½ x 4 + 2= 3 + 6 +2 + 2= 13 %Percentage error in P = 13 %Value of P is given as 3.763.By rounding off the given value to the first decimal place, we get P = 3.8.

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Answer 2

$\underline{\underline{\mathcal{ SOLUTION:-}}}$

$\rm \: P= \: \frac{a^{3} b^{2} }{(\sqrt{cd} ) \: }}$

$\rm\: \frac{\Delta \: P}{P} \: = \frac{ 3 \Delta \: a}{a}+ \frac{2 \: \Delta \: b}{b}+ \frac{1 \Delta \: c}{2 \: c}+ \frac{\Delta \: d}{d}$

$\rm \bigg(\frac{\Delta \: P}{P} \times \: 100 \bigg)\%= \bigg(3 \times \: \frac{\Delta \: a}{a} \times \: 100 + 2 \: \times \frac{\Delta \: b}{b}\times \: 100+ \frac{1}{2} \times \frac{\Delta \: c}{c}\times \: 100 + \frac{\Delta \: d}{d}\times \: 100 \bigg)\%$$\rm =3 \times 1 + 2 \times 3 + \frac{1}{2} \times 4 +2$

$\rm =3+6+2+2 = 13\%$

$\tt \: Percentage \: error \: in \: P=13\%$

$\tt \: Value \: of \: P \: is \: given \: as \: 3.763$$\heat$

$\tt \: By \: rounding \: off \: the \: given \: value \: to\\\tt \: the \: first \: decimal \: we \: get \: P=3.8.$