In an experiment, the percentage errors that occurred in the measurement of physical quantities $A$, $B$, $C$,

Question

In an experiment, the percentage errors that occurred in the measurement of physical quantities $A$, $B$, $C$, and $D$ are $1\%$, $2\%$, $3\%$, and $4\%$ respectively. Then, the maximum percentage of error in the measurement of $X$, where $X = \frac{A^2 B^{1/2}}{C^{1/3} D^3}$, will be:

Accepted Answer

1. **Identify the formula for the physical quantity:** The quantity $X$ is given by $X = \frac{A^2 B^{1/2}}{C^{1/3} D^3}$.
2. **Apply the rule for maximum relative error:** For a quantity expressed as a product or quotient of other quantities, the maximum relative error is the sum of the individual relative errors multiplied by their respective powers. Thus, the percentage error is:
 $\frac{\Delta X}{X} 	imes 100 = 2\left(\frac{\Delta A}{A} 	imes 100ight) + \frac{1}{2}\left(\frac{\Delta B}{B} 	imes 100ight) + \frac{1}{3}\left(\frac{\Delta C}{C} 	imes 100ight) + 3\left(\frac{\Delta D}{D} 	imes 100ight)$
3. **Substitute the given percentage errors:** 
 $\% 	ext{ error in } X = 2(1\%) + \frac{1}{2}(2\%) + \frac{1}{3}(3\%) + 3(4\%)$
 $\% 	ext{ error in } X = 2\% + 1\% + 1\% + 12\% = 16\%$.

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