A physical quantity X is related to three observable a,b and c as `X=(b^(2)sqrt(a))/c`. The errors of measuremnts in a,b and c are 4%, 3% and 2% respectively. What is the percentage error in the quality X?

Explanation

To find the percentage error in the physical quantity \( X \) defined by the equation:

\[ X = \frac{b^2 \sqrt{a}}{c} \]

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

\[ \text{Percentage Error in } X = \sqrt{\left( \frac{\partial X}{\partial a} \cdot \frac{\Delta a}{a} \right)^2 + \left( \frac{\partial X}{\partial b} \cdot \frac{\Delta b}{b} \right)^2 + \left( \frac{\partial X}{\partial c} \cdot \frac{\Delta c}{c} \right)^2} \]

Where \( \Delta a, \Delta b, \Delta c \) are the absolute errors in \( a, b, c \) respectively.

### Step 1: Identify the relationship and the variables
Given:
- \( X = \frac{b^2 \sqrt{a}}{c} \)
- Errors in measurements:
- \( \Delta a = 4\% \)
- \( \Delta b = 3\% \)
- \( \Delta c = 2\% \)

### Step 2: Calculate the contribution of each variable to the error in \( X \)
1. **For \( a \)**:
- The contribution to the error from \( a \) is given by:
\[ \text{Error from } a = \frac{1}{2} \cdot \frac{\Delta a}{a} \cdot 100\% = \frac{1}{2} \cdot 4\% = 2\% \]

2. **For \( b \)**:
- The contribution to the error from \( b \) is given by:
\[ \text{Error from } b = 2 \cdot \frac{\Delta b}{b} \cdot 100\% = 2 \cdot 3\% = 6\% \]

3. **For \( c \)**:
- The contribution to the error from \( c \) is given by:
\[ \text{Error from } c = -\frac{\Delta c}{c} \cdot 100\% = -2\% \]

### Step 3: Combine the errors
Using the formula for combining percentage errors, we add the contributions:
\[ \text{Total Percentage Error in } X = \text{Error from } a + \text{Error from } b + \text{Error from } c \]
Substituting the values:
\[ \text{Total Percentage Error in } X = 2\% + 6\% + 2\% = 10\% \]

### Final Answer
The percentage error in the quantity \( X \) is **10%**.

---