A physical quantity x is calculated from the relation `x = a^(3)b^(2)//sqrt(cd)`. Calculate percentage error in x, if a, b, c and d are measured respectively with an error of 1 %, 3 %, 4% and 2%.

Explanation

To calculate the percentage error in the physical quantity \( x \) given by the relation

\[ x = \frac{a^3 b^2}{\sqrt{cd}}, \]

we need to consider the errors in the measurements of \( a \), \( b \), \( c \), and \( d \). The errors in these quantities are given as follows:

- \( a \) has a percentage error of \( 1\% \)
- \( b \) has a percentage error of \( 3\% \)
- \( c \) has a percentage error of \( 4\% \)
- \( d \) has a percentage error of \( 2\% \)

### Step 1: Identify the contributions to the error in \( x \)

The formula for \( x \) involves powers of \( a \), \( b \), \( c \), and \( d \). The general rule for calculating the percentage error in a product or quotient is that the percentage errors are added, weighted by their respective powers.

For the given relation:

- The contribution from \( a \) is \( 3 \times \text{(percentage error in } a) \)
- The contribution from \( b \) is \( 2 \times \text{(percentage error in } b) \)
- The contribution from \( c \) is \( \frac{1}{2} \times \text{(percentage error in } c) \)
- The contribution from \( d \) is \( \frac{1}{2} \times \text{(percentage error in } d) \)

### Step 2: Write the expression for the total percentage error in \( x \)

The total percentage error in \( x \) can be expressed as:

\[ \text{Percentage error in } x = 3 \times \text{(percentage error in } a) + 2 \times \text{(percentage error in } b) + \frac{1}{2} \times \text{(percentage error in } c) + \frac{1}{2} \times \text{(percentage error in } d) \]

### Step 3: Substitute the values of the percentage errors

Now, substituting the given percentage errors:

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

### Step 4: Calculate each term

Calculating each term:

1. \( 3 \times 1\% = 3\% \)
2. \( 2 \times 3\% = 6\% \)
3. \( \frac{1}{2} \times 4\% = 2\% \)
4. \( \frac{1}{2} \times 2\% = 1\% \)

### Step 5: Sum all contributions

Now, summing all these contributions:

\[ \text{Percentage error in } x = 3\% + 6\% + 2\% + 1\% = 12\% \]

### Final Answer

Thus, the percentage error in \( x \) is

\[ \boxed{12\%}. \]