An experiment from `X = (a^(1//2) b^(2))/( c^(3))` . If the percentage errors in ` a, b , and c are +- 1% , +- 3% , and +- 2%` , respectively , then the percentage error in `X` can be

Options

A. `+- 12.5%`
B. `+-7%`
C. `+-1%`
D. `+-4%`

Explanation

To find the percentage error in \( X \) given the expression \( X = \frac{a^{1/2} b^{2}}{c^{3}} \) and the percentage errors in \( a, b, \) and \( c \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the formula for percentage error in a function**:
The percentage error in a function \( X \) that depends on multiple variables can be calculated using the formula:
\[ \frac{\Delta X}{X} \times 100 = \left( n_1 \frac{\Delta A}{A} + n_2 \frac{\Delta B}{B} + n_3 \frac{\Delta C}{C} \right) \times 100 \]
where \( n_1, n_2, n_3 \) are the powers of \( a, b, c \) in the expression for \( X \).

2. **Substitute the powers and the percentage errors**:
For our case:
- \( n_1 = \frac{1}{2} \) (for \( a \))
- \( n_2 = 2 \) (for \( b \))
- \( n_3 = -3 \) (for \( c \))

The percentage errors given are:
- \( \frac{\Delta A}{A} = 1\% \)
- \( \frac{\Delta B}{B} = 3\% \)
- \( \frac{\Delta C}{C} = 2\% \)

3. **Plug in the values**:
Now substituting these values into the formula:
\[ \frac{\Delta X}{X} \times 100 = \left( \frac{1}{2} \times 1 + 2 \times 3 - 3 \times 2 \right) \]

4. **Calculate each term**:
- For \( a \): \( \frac{1}{2} \times 1 = 0.5\% \)
- For \( b \): \( 2 \times 3 = 6\% \)
- For \( c \): \( -3 \times 2 = -6\% \)

5. **Combine the results**:
Now, add these results together:
\[ 0.5\% + 6\% - 6\% = 0.5\% \]

6. **Final calculation of percentage error in \( X \)**:
Since we are calculating the total percentage error in \( X \):
\[ \text{Percentage error in } X = 0.5\% + 6\% + 6\% = 12.5\% \]

### Final Result:
The percentage error in \( X \) can be \( \pm 12.5\% \).

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