Question

A quantity \(Q\) is formulated \(X^{-2}Y^{+\frac{3}{2}}Z^{-\frac{2}{5}}\). \(X\), \(Y\) 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. \(0.6\)
• B. \(0.7\)
• C. \(0.8\)
• D. \(0.1\)

Hint:

For a product of powers, the maximum fractional error is the sum of each exponent magnitude times the fractional error of that parameter.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
For \(Q=X^{a}Y^{b}Z^{c}\), the maximum fractional error is \(\frac{\Delta Q}{Q}=|a|\frac{\Delta X}{X}+|b|\frac{\Delta Y}{Y}+|c|\frac{\Delta Z}{Z}\).
Signs of the exponents do not matter because errors add up in the worst case.

Step 2: Identify the values
Exponents: \(a=-2\), \(b=\frac{3}{2}\), \(c=-\frac{2}{5}\).
Fractional errors: \(0.1\), \(0.2\), \(0.5\).

Step 3: Calculation
\(\frac{\Delta Q}{Q}=2(0.1)+\frac{3}{2}(0.2)+\frac{2}{5}(0.5)\)
\(=0.2+0.3+0.2=0.7\)

Step 4: Compare the options
The value \(0.7\) matches option (B).

Final Answer:
The maximum fractional error of Q is 0.7. \[ \boxed{\text{(B) }0.7} \]