Question

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

• A. \((\frac{3}{13})\%\)
• B. \(16\%\)
• C. \(-10\%\)
• D. \(10\%\)

Answer 1

The Correct Option is B

To find the maximum percentage of error in the measurement of \( X \), where \( X =\frac{A^2B^{1/2}}{C^{1/3}D^3} \), we first need to understand how errors propagate in a formula involving multiple variables.

When dealing with functions of multiple variables, the percentage error in the result is the sum of the percentage errors contributed by each variable, each multiplied by the absolute value of the power to which that variable is raised in the expression.

The expression for \( X \) is given by:

X = \frac{A^2B^{1/2}}{C^{1/3}D^3}

We can now write the formula for the percentage error in \( X \) as follows:

\(\text{Percentage error in } X = 2 \cdot (\text{Percentage error in } A) + \frac{1}{2} \cdot (\text{Percentage error in } B) + \frac{1}{3} \cdot (\text{Percentage error in } C) + 3 \cdot (\text{Percentage error in } D)\)

Substituting the given percentage errors:

• 1\%\) for \(A
• 2\%\) for \(B
• 3\%\) for \(C
• 4\%\) for \(D

We substitute the values into the percentage error formula:

\(\text{Percentage error in } X = 2 \times 1 + \frac{1}{2} \times 2 + \frac{1}{3} \times 3 + 3 \times 4\)

Calculating each term:

• 2 \times 1\% = 2\%
• \frac{1}{2} \times 2\% = 1\%
• \frac{1}{3} \times 3\% = 1\%
• 3 \times 4\% = 12\%

Summing up the percentage errors:

2\% + 1\% + 1\% + 12\% = 16\%

Hence, the maximum percentage of error in the measurement of \( X \) is 16\%.