A physical quantity `x` is calculated from ` x = ab^(2)//sqrt(c )`. Calculate the percentage error in measuring `x` when the percentage errors in measuring a , b , and c are 4 , 2 , and 3%, respectively .

Options

A. `7%`
B. `9%`
C. `11%`
D. `9.5%`

Explanation

To calculate the percentage error in measuring the physical quantity \( x \), given by the formula:

\[ x = \frac{ab^2}{\sqrt{c}} \]

we need to consider the percentage errors in the measurements of \( a \), \( b \), and \( c \).

### Step 1: Identify the formula for percentage error
The formula for the percentage error in a product or quotient of quantities is given by the sum of the relative errors of each quantity, adjusted for their respective powers in the formula.

For our case, the formula for \( x \) can be expressed in terms of its components:

\[ \frac{\Delta x}{x} \text{ (percentage error in } x\text{)} = \frac{\Delta a}{a} + 2 \frac{\Delta b}{b} - \frac{1}{2} \frac{\Delta c}{c} \]

### Step 2: Substitute the given percentage errors
We are given the following percentage errors:
- Percentage error in \( a \) = 4%
- Percentage error in \( b \) = 2%
- Percentage error in \( c \) = 3%

Now we can substitute these values into our formula:

\[ \frac{\Delta x}{x} = 4\% + 2 \times 2\% - \frac{1}{2} \times 3\% \]

### Step 3: Calculate the contributions
Now we calculate each term:
- The contribution from \( a \) is \( 4\% \).
- The contribution from \( b \) is \( 2 \times 2\% = 4\% \).
- The contribution from \( c \) is \( -\frac{1}{2} \times 3\% = -1.5\% \).

### Step 4: Sum the contributions
Now we sum these contributions to find the total percentage error in \( x \):

\[ \frac{\Delta x}{x} = 4\% + 4\% - 1.5\% = 6.5\% \]

### Step 5: Final Calculation
Thus, the total percentage error in measuring \( x \) is:

\[ \frac{\Delta x}{x} = 4\% + 4\% - 1.5\% = 6.5\% \]

### Conclusion
The percentage error in measuring \( x \) is **6.5%**.

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