Question

A physical quantity \(P\) is related to four observables \(a\), \(b\), \(c\), and \(d\) as \[ P=\frac{\sqrt{ab}\cdot d^{\alpha}}{\sqrt{c}} \] (\(\alpha\) is a constant). The percentage errors in \(a\), \(b\), \(c\), and \(d\) are \(0.5\%\) each. If the percentage error in \(P\) is \(2\%\), then \(\alpha\) is

• A. \(\frac{5}{2}\)
• B. \(\frac{2}{5}\)
• C. \(\frac{3}{4}\)
• D. \(\frac{3}{2}\)

Hint:

For products and quotients, \[ Q=x^ay^bz^c, \] the maximum percentage error is obtained by adding the absolute values of the powers multiplied by their respective percentage errors: \[ \%\Delta Q = |a|\%\Delta x + |b|\%\Delta y + |c|\%\Delta z. \]

Answer 1

The Correct Option is A

Step 1: Write the expression in power form.
Given, \[ P=\frac{\sqrt{ab}\cdot d^{\alpha}}{\sqrt{c}} \] This can be written as \[ P=a^{1/2}b^{1/2}c^{-1/2}d^{\alpha}. \]

Step 2: Use the formula for maximum percentage error.
For a quantity \[ Q=x^my^nz^p, \] the maximum percentage error is \[ \frac{\Delta Q}{Q}\times 100 = \left( |m|\frac{\Delta x}{x} + |n|\frac{\Delta y}{y} + |p|\frac{\Delta z}{z} \right)\times 100. \] Applying this to \(P\), \[ \frac{\Delta P}{P}\times 100 = \left( \frac{1}{2}\frac{\Delta a}{a} + \frac{1}{2}\frac{\Delta b}{b} + \frac{1}{2}\frac{\Delta c}{c} + \alpha\frac{\Delta d}{d} \right)\times 100. \]

Step 3: Substitute the given percentage errors.
The percentage errors in \[ a,b,c,d \] are each \[ 0.5\%. \] Hence, \[ \%\text{ error in }P = \left( \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\alpha \right)(0.5). \] Given percentage error in \(P\) is \[ 2\%. \] Therefore, \[ \left(\frac{3}{2}+\alpha\right)(0.5)=2. \]

Step 4: Solve for \(\alpha\).
Multiplying both sides by \(2\), \[ \frac{3}{2}+\alpha=4. \] Hence, \[ \alpha=4-\frac{3}{2}. \] \[ \alpha=\frac{8-3}{2}. \] \[ \alpha=\frac{5}{2}. \]

Step 5: Final conclusion.
Therefore, \[ \boxed{\alpha=\frac{5}{2}} \] Hence, the correct option is \[ \boxed{(1)} \]