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A physical quantity $X$ is related to four measurable quantities $a, b, c$ and $d$ follows $X=a^2 b^3 c^{5 / 2} d^{-2}$. The perecentage error in the measurement of $a, b, c$ and $d$ are $1 \%, 2 \%, 3 \%$ and $4 \%$, respectively. What is the percentage error in quantity $X$ ? If the value of $X$ calculated on the basis of the above relation is $2.763$, to what value should you round off the result.
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As we know that, the formula of percentage error in quantity $X$ is $\frac{\Delta X}{X} \times 100$
As given that physical quantity is
$$
X=a^2 b^3 c^{5 / 2} d^{-2}
$$
Maximum percentage error in $X$ is
$$
\begin{aligned}
\frac{\Delta X}{X} \times 100=& \pm\left[2\left(\frac{\Delta a}{a} \times 100\right)+3\left(\frac{\Delta b}{b} \times 100\right)\right.\\
&\left.+\frac{5}{2}\left(\frac{\Delta c}{c} \times 100\right)+2\left(\frac{\Delta d}{d} \times 100\right)\right] \\
=& \pm\left[2(1)+3(2)+\frac{5}{2}(3)+2(4)\right] \times \frac{100}{100} \\
=& \pm\left[2+6+\frac{15}{2}+8\right]=\pm 23.5 \%
\end{aligned}
$$
So, percentage error in quantity $X=\pm 23.5 \%$
Mean absolute error in $X=\pm \frac{23.5}{100}=\pm 0.235$ $=\pm 0.24$ (rounding-off upto two significant digits) Then the calculated value of $X$ should be rounding off upto two significant digits.
So, $X=2.768$ or $X=2.8$.
As given that physical quantity is
$$
X=a^2 b^3 c^{5 / 2} d^{-2}
$$
Maximum percentage error in $X$ is
$$
\begin{aligned}
\frac{\Delta X}{X} \times 100=& \pm\left[2\left(\frac{\Delta a}{a} \times 100\right)+3\left(\frac{\Delta b}{b} \times 100\right)\right.\\
&\left.+\frac{5}{2}\left(\frac{\Delta c}{c} \times 100\right)+2\left(\frac{\Delta d}{d} \times 100\right)\right] \\
=& \pm\left[2(1)+3(2)+\frac{5}{2}(3)+2(4)\right] \times \frac{100}{100} \\
=& \pm\left[2+6+\frac{15}{2}+8\right]=\pm 23.5 \%
\end{aligned}
$$
So, percentage error in quantity $X=\pm 23.5 \%$
Mean absolute error in $X=\pm \frac{23.5}{100}=\pm 0.235$ $=\pm 0.24$ (rounding-off upto two significant digits) Then the calculated value of $X$ should be rounding off upto two significant digits.
So, $X=2.768$ or $X=2.8$.
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