A physical quantity ' \(X\) ' is related to four quantities as \(X = a^2 b^3 c^{5/2} d^{-2}\). The percentage error in 'a', 'b', 'c' and 'd' are \(1%\), \(2%\), \(2%\) and \(4%\) respectively. The percentage error in ' \(X\) ' is.
Show Hint
Always add the products of power and percentage error, regardless of whether the term is in the numerator or denominator.
Step 1: Understanding the Question:
The question asks for the propagation of percentage errors in a derived physical quantity based on the errors in its constituent base quantities. Step 2: Key Formula or Approach:
For a formula \(X = a^p b^q c^r d^s\), the maximum percentage error in \(X\) is:
\[ \frac{\Delta X}{X} \times 100 = \left| p \frac{\Delta a}{a} \right| \times 100 + \left| q \frac{\Delta b}{b} \right| \times 100 + \left| r \frac{\Delta c}{c} \right| \times 100 + \left| s \frac{\Delta d}{d} \right| \times 100 \] Step 3: Detailed Explanation:
Given \(X = a^{2} b^{3} c^{5/2} d^{-2}\).
Percentage errors are:
\(\frac{\Delta a}{a} \times 100 = 1%\)
\(\frac{\Delta b}{b} \times 100 = 2%\)
\(\frac{\Delta c}{c} \times 100 = 2%\)
\(\frac{\Delta d}{d} \times 100 = 4%\)
The percentage error in \(X\) is:
\[ % \text{ error in } X = 2(% \text{ in } a) + 3(% \text{ in } b) + \frac{5}{2}(% \text{ in } c) + 2(% \text{ in } d) \]
Note: Errors always add up, so the negative exponent of \(d\) is taken as its absolute value.
\[ % \text{ error in } X = 2(1) + 3(2) + \frac{5}{2}(2) + 2(4) \]
\[ % \text{ error in } X = 2 + 6 + 5 + 8 \]
\[ % \text{ error in } X = 21% \] Step 4: Final Answer:
The percentage error in \(X\) is \(21%\).