A quantity $ Q $ is formulated as $ Q = X^{-2} Y^{3/2} Z^{-2/5} $. $ X $, $ Y $, and $ Z $ are independent parameters which have fractional errors of 0.1, 0.2, and 0.5, respectively in measurement. The maximum fractional error of $ Q $ is:
Show Hint
- 0.7
- 0.1
- 0.8
- 0.6
The Correct Option is A
Approach Solution - 1
To determine the maximum fractional error in the quantity \(Q\), we will need to consider how the errors in parameters \(X\), \(Y\), and \(Z\) propagate through the equation. The given quantity is:
\(Q = X^{-2} Y^{3/2} Z^{-2/5}\)
The fractional error of a product or quotient can be found by taking the partial derivative with respect to each variable, multiplying by the corresponding fractional error, and summing the absolute values of these terms:
The general formula for fractional error in a multiplicative relationship \(Q = X^a Y^b Z^c\) is:
\(\frac{\Delta Q}{Q} = |a| \frac{\Delta X}{X} + |b| \frac{\Delta Y}{Y} + |c| \frac{\Delta Z}{Z}\)
For the given function \(Q = X^{-2} Y^{3/2} Z^{-2/5}\), the exponents are \(a = -2\), \(b = \frac{3}{2}\), and \(c = -\frac{2}{5}\).
Given the fractional errors:
- \(\frac{\Delta X}{X} = 0.1\)
- \(\frac{\Delta Y}{Y} = 0.2\)
- \(\frac{\Delta Z}{Z} = 0.5\)
Using the formula for fractional error, we find:
\(\frac{\Delta Q}{Q} = |-2| \times 0.1 + \left|\frac{3}{2}\right| \times 0.2 + \left|-\frac{2}{5}\right| \times 0.5\)
Calculating each term:
- \(|-2| \times 0.1 = 0.2\)
- \(\left|\frac{3}{2}\right| \times 0.2 = 0.3\)
- \frac{\Delta Q}{Q} = 0.2 + 0.3 + 0.2 = 0.7
Therefore, the maximum fractional error of \(Q\) is 0.7.
Approach Solution -2
For a function of several variables, the fractional error is given by: \[ \frac{\Delta Q}{Q} = \left| \frac{\partial Q}{\partial X} \frac{\Delta X}{X} \right| + \left| \frac{\partial Q}{\partial Y} \frac{\Delta Y}{Y} \right| + \left| \frac{\partial Q}{\partial Z} \frac{\Delta Z}{Z} \right| \] For the given function, we calculate the partial derivatives with respect to \( X \), \( Y \), and \( Z \): \[ \frac{\partial Q}{\partial X} = -2 X^{-3}, \quad \frac{\partial Q}{\partial Y} = \frac{3}{2} Y^{1/2}, \quad \frac{\partial Q}{\partial Z} = -\frac{2}{5} Z^{-7/5} \] Now, applying the error propagation formula, the fractional error in \( Q \) is: \[ \frac{\Delta Q}{Q} = 2 \times \frac{\Delta X}{X} + \frac{3}{2} \times \frac{\Delta Y}{Y} + \frac{2}{5} \times \frac{\Delta Z}{Z} \] Substitute the given fractional errors for \( X \), \( Y \), and \( Z \): \[ \frac{\Delta Q}{Q} = 2 \times 0.1 + \frac{3}{2} \times 0.2 + \frac{2}{5} \times 0.5 \] Simplifying: \[ \frac{\Delta Q}{Q} = 0.2 + 0.3 + 0.2 = 0.7 \] Thus, the maximum fractional error in \( Q \) is \( 0.7 \). Therefore, the correct answer is Option (1).
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