Let `x={:[(a^(2)b^(2))/(c)]:}` be the physical quantity. If the percentage error in the measurement of physical quantities a,b, and c is 2,3 and 4 per cent respectively, then percentage error in the measurement of x is

Options

A. 0.07
B. 0.14
C. 0.21
D. 0.28

Answer

To solve the problem, we need to find the percentage error in the physical quantity \( x \), which is defined as:

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

Given the percentage errors in the measurements of \( a \), \( b \), and \( c \):
- Percentage error in \( a \) = 2%
- Percentage error in \( b \) = 3%
- Percentage error in \( c \) = 4%

### Step 1: Understanding the formula for percentage error
The formula for the percentage error in a quantity that is a function of other quantities is given by:

\[ \text{Percentage error in } x = \left( \text{Percentage error in } a + \text{Percentage error in } b + \text{Percentage error in } c \right) \]

However, since \( x \) involves \( a^2 \) and \( b^2 \), we need to consider the powers of \( a \) and \( b \) in the formula.

### Step 2: Applying the formula for powers
The percentage error for a product or quotient of quantities is calculated as follows:
- For \( a^2 \), the contribution to the percentage error is \( 2 \times \text{Percentage error in } a \).
- For \( b^2 \), the contribution to the percentage error is \( 2 \times \text{Percentage error in } b \).
- For \( c \), since it is in the denominator, the contribution is \( -\text{Percentage error in } c \).

So, we can write:

\[ \text{Percentage error in } x = 2 \times \text{Percentage error in } a + 2 \times \text{Percentage error in } b + \text{Percentage error in } c \]

### Step 3: Substituting the values
Now substituting the given values into the formula:

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

Calculating each term:

\[ = 4\% + 6\% + 4\% \]

### Step 4: Adding the contributions
Now, we add these contributions together:

\[ = 4\% + 6\% + 4\% = 14\% \]

### Conclusion
Thus, the percentage error in the measurement of \( x \) is:

\[ \text{Percentage error in } x = 14\% \]

### Final Answer
The final answer is 14%.

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