A physical quantity P is described by the relation P=a^(1/2)b^(2)c^(3)d^(-4). If the relative errors in the measurements a, b, c and d respectively, are 2%, 1%, 3%, and 5%, then the relative error in P will be :

Options

A. 25%
B. 12%
C. 8%
D. 32%

Answer

To find the relative error in the physical quantity \( P \) given by the relation \( P = a^{1/2} b^{2} c^{3} d^{-4} \), we can follow these steps:

### Step 1: Write the formula for relative error
The relative error in \( P \) can be expressed in terms of the relative errors in \( a \), \( b \), \( c \), and \( d \). The formula for relative error in a product or quotient of quantities is given by:

\[ \frac{\Delta P}{P} = \frac{1}{2} \frac{\Delta a}{a} + 2 \frac{\Delta b}{b} + 3 \frac{\Delta c}{c} - 4 \frac{\Delta d}{d} \]

### Step 2: Identify the relative errors
From the problem, we have the following relative errors:
- Relative error in \( a \) (\( \frac{\Delta a}{a} \)): 2% or 0.02
- Relative error in \( b \) (\( \frac{\Delta b}{b} \)): 1% or 0.01
- Relative error in \( c \) (\( \frac{\Delta c}{c} \)): 3% or 0.03
- Relative error in \( d \) (\( \frac{\Delta d}{d} \)): 5% or 0.05

### Step 3: Substitute the values into the formula
Now, we substitute the relative errors into the formula:

\[ \frac{\Delta P}{P} = \frac{1}{2} \times 0.02 + 2 \times 0.01 + 3 \times 0.03 - 4 \times 0.05 \]

### Step 4: Calculate each term
Calculating each term step-by-step:

1. \( \frac{1}{2} \times 0.02 = 0.01 \)
2. \( 2 \times 0.01 = 0.02 \)
3. \( 3 \times 0.03 = 0.09 \)
4. \( -4 \times 0.05 = -0.20 \)

### Step 5: Combine the results
Now we combine all the calculated terms:

\[ \frac{\Delta P}{P} = 0.01 + 0.02 + 0.09 - 0.20 \]

Calculating this gives:

\[ \frac{\Delta P}{P} = 0.01 + 0.02 + 0.09 - 0.20 = -0.08 \]

### Step 6: Convert to percentage
To find the relative error in percentage, we multiply by 100:

\[ \text{Relative error in } P = -0.08 \times 100 = -8\% \]

### Final Answer
The relative error in \( P \) is \( -8\% \).

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