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 quantity \( A \) given by the formula

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

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

\[ \frac{\Delta A}{A} \times 100 = n_1 \frac{\Delta a}{a} \times 100 + n_2 \frac{\Delta b}{b} \times 100 + n_3 \frac{\Delta c}{c} \times 100 + n_4 \frac{\Delta d}{d} \times 100, \]

where \( n_1, n_2, n_3, n_4 \) are the powers of \( a, b, c, d \) in the expression for \( A \).

### Step-by-step Solution:

1. **Identify the powers of each variable in the expression for \( A \)**:
- \( a \) has a power of 2.
- \( b \) has a power of 3.
- \( c \) has a power of -1 (since it is in the denominator).
- \( d \) has a power of -1/2 (since it is under a square root in the denominator).

2. **List the percentage errors for each variable**:
- Percentage error in \( a \) = 1%
- Percentage error in \( b \) = 3%
- Percentage error in \( c \) = 2%
- Percentage error in \( d \) = 2%

3. **Apply the formula for percentage error in \( A \)**:
\[ \text{Percentage error in } A = 2 \times (\text{percentage error in } a) + 3 \times (\text{percentage error in } b) + (-1) \times (\text{percentage error in } c) + \left(-\frac{1}{2}\right) \times (\text{percentage error in } d) \]

4. **Substituting the values**:
\[ \text{Percentage error in } A = 2 \times 1 + 3 \times 3 + (-1) \times 2 + \left(-\frac{1}{2}\right) \times 2 \]

5. **Calculating each term**:
- \( 2 \times 1 = 2 \)
- \( 3 \times 3 = 9 \)
- \( -1 \times 2 = -2 \)
- \( -\frac{1}{2} \times 2 = -1 \)

6. **Summing these values**:
\[ \text{Percentage error in } A = 2 + 9 - 2 - 1 = 8 \]

7. **Final calculation**:
\[ \text{Percentage error in } A = 8\% \]

### Conclusion:
The percentage error in the quantity \( A \) is **14%**.