A physical quantity A is related to four observable a,b,c and d as follows, `A=(a^2b^3)/(csqrtd)`, the percentage errors of measurement is a,b,c and d,are `1%`,`3%`,`2%` and `2%` respectively. What is the percentage error in the quantity A?

Options

A. `12%`
B. `7%`
C. `5%`
D. `14%`

Explanation

To find the percentage error in the physical quantity \( A \) given by the formula:

\[ A = \frac{a^2 b^3}{c \sqrt{d}} \]

we will use the formula for calculating the percentage error in a quantity defined in terms of other quantities. The formula for the percentage error in a quantity \( Z \) defined as:

\[ Z = A^m B^n C^p D^q \]

is given by:

\[ \text{Percentage Error in } Z = |m| \cdot \text{Percentage Error in } A + |n| \cdot \text{Percentage Error in } B + |p| \cdot \text{Percentage Error in } C + |q| \cdot \text{Percentage Error in } D \]

### Step 1: Identify the exponents and their corresponding percentage errors

From the formula for \( A \):

- \( a \) has an exponent of 2, and its percentage error is \( 1\% \)
- \( b \) has an exponent of 3, and its percentage error is \( 3\% \)
- \( c \) has an exponent of -1 (since it is in the denominator), and its percentage error is \( 2\% \)
- \( d \) has an exponent of -1/2 (since it is under a square root in the denominator), and its percentage error is \( 2\% \)

### Step 2: Apply the formula for percentage error

Using the formula, we calculate the total percentage error in \( A \):

\[ \text{Percentage Error in } A = |2| \cdot 1\% + |3| \cdot 3\% + |-1| \cdot 2\% + \left| -\frac{1}{2} \right| \cdot 2\% \]

Substituting the values:

\[ = 2 \cdot 1 + 3 \cdot 3 + 1 \cdot 2 + \frac{1}{2} \cdot 2 \]

### Step 3: Calculate each term

Calculating each term step-by-step:

1. \( 2 \cdot 1 = 2 \)
2. \( 3 \cdot 3 = 9 \)
3. \( 1 \cdot 2 = 2 \)
4. \( \frac{1}{2} \cdot 2 = 1 \)

### Step 4: Sum the contributions

Now, summing these contributions:

\[ \text{Total Percentage Error} = 2 + 9 + 2 + 1 = 14\% \]

### Conclusion

Thus, the percentage error in the quantity \( A \) is \( 14\% \).

### Final Answer

The percentage error in the quantity \( A \) is \( 14\% \).

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