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

The Correct Option is C

To find the percentage error in the physical quantity \( Q \) given by the relation \(Q = \frac{a^4 b^3}{c^2}\), we use the formula for percentage error in a derived quantity: 

• The percentage error in a physical quantity is obtained by summing the relative percentage errors of each variable multiplied by their respective powers.

Let's analyze each of the terms individually:

1. The power of \( a \) in the expression is \( 4 \). Since the percentage error in \( a \) is \( 3\% \), the contribution of \( a \) to the percentage error in \( Q \) will be \( 4 \times 3\% = 12\% \).
2. The power of \( b \) is \( 3 \). Thus, the contribution of \( b \) to the percentage error in \( Q \) will be \( 3 \times 4\% = 12\% \).
3. The power of \( c \) is \(-2\) (because it is in the denominator, we take it as a negative value). Therefore, the contribution of \( c \) to the percentage error in \( Q \) will be \( -2 \times 5\% = -10\% \).

Therefore, the total percentage error in \( Q \) is:

Percentage error in \( Q \)	= \( 12\% + 12\% + 10\% \)
	= \( 12\% + 12\% + 10\% = 34\% \)

Hence, the percentage error in \( Q \) is \(34\%.\)

Therefore, the correct answer is 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%