Question

A physical quantity \(x\) is related as \(x = \sqrt{a} b^2c^3d^{-4}\). Relative errors in the quantities \(a\), \(b\), \(c\) and \(d\) are 2%, 1%, 3% and 4% respectively. Relative error in \(x\) will be

• A. 30%
• B. 28%
• C. 24%
• D. 18%

Hint:

Powers multiply the relative errors; add all the contributions.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
When quantities are multiplied or divided, their relative errors add. A power \(n\) on a quantity multiplies its relative error by \(n\).

Step 2: Key Formula or Approach
For \(x=a^{1/2}b^2c^3d^{-4}\):
\[ \frac{\Delta x}{x}=\frac12\frac{\Delta a}{a}+2\frac{\Delta b}{b}+3\frac{\Delta c}{c}+4\frac{\Delta d}{d} \]
The negative power on \(d\) only changes the sign inside the formula, but errors are added in the worst case, so we use \(|{-4}|=4\).

Step 3: Substitute
\[ \frac{\Delta x}{x}=\frac12(2)+2(1)+3(3)+4(4)\ \% \]
\[ =1+2+9+16=28\ \% \]

Step 4: Check the options
A value of 30% would come from forgetting the half on \(a\) and using 2 instead of 1. The value 24% would come from losing the 4 in the \(d\) term. The result is 28%, option (B).

Final Answer:
The maximum relative error is 1 + 2 + 9 + 16 = 28 percent, option (B). \[ \boxed{28\%} \]