Question

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:

• A. 0.7
• B. 0.1
• C. 0.8
• D. 0.6

Hint:

When dealing with propagation of errors, remember to apply the error propagation formula, which involves the sum of the products of the partial derivatives and the corresponding fractional errors of each variable.

Answer 1

The Correct Option is A

The formula for \( Q \) is given as: \[ Q = X^{-2} Y^{3/2} Z^{-2/5} \] To find the maximum fractional error of \( Q \), we use the formula for propagation of errors.
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).