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 calculate the percentage error in the physical quantity \( Q \), defined by \(Q = \frac{a^4 b^3}{c^2}\), we sum the relative percentage errors of its constituent variables, weighted by their respective powers.

• The percentage error in a derived physical quantity is the sum of the products of the power of each variable and its percentage error.

Analyzing each term:

1. The term \( a \) is raised to the power of \( 4 \). With a percentage error of \( 3\% \) for \( a \), its contribution to the total percentage error in \( Q \) is \( 4 \times 3\% = 12\% \).
2. The term \( b \) is raised to the power of \( 3 \). Its contribution to the percentage error in \( Q \) is \( 3 \times 4\% = 12\% \).
3. The term \( c \) is raised to the power of \(-2\) (due to its presence in the denominator). Its contribution to the percentage error in \( Q \) is \( -2 \times 5\% = -10\% \).

The total percentage error in \( Q \) is computed as follows:

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

Consequently, the percentage error in \( Q \) is \(34\%\). Therefore, the correct answer is 34%.