In an experiment four quantities a,b,c and d are measure with percentage error `1% , 2% , 3%`,and `4%` respectively quantity is P is calculate as follow
`P = (a^(3)b^(2))/(cd) %` error in `P` is

Options

A. `14%`
B. `10%`
C. `7%`
D. `4%`

Explanation

To find the percentage error in the quantity \( P \) defined as

\[ P = \frac{a^3 b^2}{cd} \]

given the percentage errors in \( a \), \( b \), \( c \), and \( d \) as \( 1\% \), \( 2\% \), \( 3\% \), and \( 4\% \) respectively, we can follow these steps:

### Step 1: Identify the formula for percentage error
The percentage error in a quantity calculated from other quantities can be determined using the following rule:
- For a product or quotient, the percentage errors are added, and for powers, the percentage error is multiplied by the exponent.

### Step 2: Write down the contributions to the percentage error
For the given formula \( P = \frac{a^3 b^2}{cd} \), we can break it down as follows:
- The contribution from \( a^3 \) is \( 3 \times \text{(percentage error in } a) \)
- The contribution from \( b^2 \) is \( 2 \times \text{(percentage error in } b) \)
- The contribution from \( c \) is \( \text{(percentage error in } c) \)
- The contribution from \( d \) is \( \text{(percentage error in } d) \)

### Step 3: Substitute the given percentage errors
Given:
- Percentage error in \( a = 1\% \)
- Percentage error in \( b = 2\% \)
- Percentage error in \( c = 3\% \)
- Percentage error in \( d = 4\% \)

Now, we can substitute these values into the formula for the total percentage error in \( P \):

\[ \text{Percentage error in } P = 3 \times 1\% + 2 \times 2\% + 3\% + 4\% \]

### Step 4: Calculate the total percentage error
Now, we perform the calculations:

\[ \text{Percentage error in } P = 3 \times 1 + 2 \times 2 + 3 + 4 \]
\[ = 3 + 4 + 3 + 4 = 14\% \]

### Final Answer
The percentage error in \( P \) is \( 14\% \).

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