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 error in a, b and c are 3%, 4% and 5% respectively. Then, the percentage error in Q is :

Options

A. `66%`
B. `43%`
C. `34%`
D. `14%`

Answer

To find the percentage error in the physical quantity \( Q \) given by the relation

\[ Q = \frac{a^4 b^3}{c^2} \]

we can use the formula for the propagation of errors. The percentage error in a quantity that is a function of several variables can be calculated using the following formula:

\[ \frac{\Delta Q}{Q} \times 100 = \left( n_a \frac{\Delta a}{a} + n_b \frac{\Delta b}{b} + n_c \frac{\Delta c}{c} \right) \times 100 \]

where \( n_a, n_b, n_c \) are the powers of \( a, b, c \) in the expression for \( Q \), and \( \Delta a, \Delta b, \Delta c \) are the absolute errors in \( a, b, c \) respectively.

### Step 1: Identify the powers of each variable
From the given relation \( Q = \frac{a^4 b^3}{c^2} \):
- The power of \( a \) is \( 4 \) (i.e., \( n_a = 4 \))
- The power of \( b \) is \( 3 \) (i.e., \( n_b = 3 \))
- The power of \( c \) is \( -2 \) (i.e., \( n_c = -2 \))

### Step 2: Write down the percentage errors
The percentage errors given are:
- For \( a \): \( \frac{\Delta a}{a} \times 100 = 3\% \) (i.e., \( \frac{\Delta a}{a} = 0.03 \))
- For \( b \): \( \frac{\Delta b}{b} \times 100 = 4\% \) (i.e., \( \frac{\Delta b}{b} = 0.04 \))
- For \( c \): \( \frac{\Delta c}{c} \times 100 = 5\% \) (i.e., \( \frac{\Delta c}{c} = 0.05 \))

### Step 3: Substitute the values into the error propagation formula
Now, substituting the values into the error propagation formula:

\[ \frac{\Delta Q}{Q} \times 100 = \left( 4 \cdot 0.03 + 3 \cdot 0.04 + (-2) \cdot 0.05 \right) \times 100 \]

### Step 4: Calculate each term
Calculating each term:
- For \( a \): \( 4 \cdot 0.03 = 0.12 \)
- For \( b \): \( 3 \cdot 0.04 = 0.12 \)
- For \( c \): \( -2 \cdot 0.05 = -0.10 \)

### Step 5: Sum the contributions
Now, summing these contributions:

\[ 0.12 + 0.12 - 0.10 = 0.14 \]

### Step 6: Convert to percentage
Now, converting to percentage:

\[ \frac{\Delta Q}{Q} \times 100 = 0.14 \times 100 = 14\% \]

### Step 7: Final calculation
Thus, the percentage error in \( Q \) is:

\[ \text{Percentage error in } Q = 14\% \]

### Conclusion
The percentage error in \( Q \) is \( 14\% \).

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