Question

A physical quantity $Q$ is found to depend on quantities $a$, $b$, $c$ by the relation$Q = \frac{a^4 b^3}{c^2}.$The percentage errors in $a$, $b$, and $c$ are 3%, 4%, and 5% respectively. Then, the percentage error in $Q$ is:

• A. 66%
• B. 43%
• C. 34%
• D. 14%

Answer 1

Answer: (C)

34%

Answer 2

Given: $Q = \frac{a^4 b^3}{c^2}$

The percentage error in $Q$ can be calculated using the rule for propagation of errors in multiplication and division.

Step 1. Calculate the fractional error in $Q$:

$\frac{\Delta Q}{Q} = 4 \frac{\Delta a}{a} + 3 \frac{\Delta b}{b} + 2 \frac{\Delta c}{c}$

Step 2. Substitute the given percentage errors:

$\frac{\Delta Q}{Q} \times 100 = 4 \left( \frac{\Delta a}{a} \times 100 \right) + 3 \left( \frac{\Delta b}{b} \times 100 \right) + 2 \left( \frac{\Delta c}{c} \times 100 \right)$

$= 4 \times 3\% + 3 \times 4\% + 2 \times 5\%$

$= 12\% + 12\% + 10\%$

Step 3. Calculate the total percentage error in $Q$:

$\text{Percentage error in } Q = 12\% + 12\% + 10\% = 34\%$

Thus, the percentage error in $Q$ is 34%.

The Correct Answer is: 34%