An experiment measure quantities x,y,z and then t is in calculate from the data as `t = (xy^(2))/(z^(2))` if perecentage error in x,y,z and are respectively `1% ,3%,2%` then percentage error in//is

Options

A. `10%`
B. `4%`
C. `7%`
D. `13%`

Explanation

To solve the problem, we need to find the percentage error in the quantity \( t \) given the formula:

\[ t = \frac{xy^2}{z^2} \]

where \( x \), \( y \), and \( z \) have percentage errors of \( 1\% \), \( 3\% \), and \( 2\% \) respectively.

### Step-by-Step Solution:

1. **Identify the formula for \( t \)**:
\[ t = \frac{xy^2}{z^2} \]

2. **Write down the percentage errors for \( x \), \( y \), and \( z \)**:
- Percentage error in \( x \) is \( 1\% \)
- Percentage error in \( y \) is \( 3\% \)
- Percentage error in \( z \) is \( 2\% \)

3. **Use the formula for propagation of errors**:
The percentage error in \( t \) can be calculated using the formula:
\[ \frac{\Delta t}{t} \times 100 = \frac{\Delta x}{x} \times 100 + 2 \times \frac{\Delta y}{y} \times 100 + 2 \times \frac{\Delta z}{z} \times 100 \]
Here, the factor of \( 2 \) is included for \( y \) and \( z \) because they are raised to the power of \( 2 \) in the formula for \( t \).

4. **Substitute the percentage errors into the formula**:
\[ \frac{\Delta t}{t} \times 100 = 1\% + 2 \times 3\% + 2 \times 2\% \]

5. **Calculate each term**:
- The first term is \( 1\% \)
- The second term is \( 2 \times 3\% = 6\% \)
- The third term is \( 2 \times 2\% = 4\% \)

6. **Sum the percentage errors**:
\[ \frac{\Delta t}{t} \times 100 = 1\% + 6\% + 4\% = 11\% \]

7. **Conclusion**:
The percentage error in \( t \) is \( 11\% \).

### Final Answer:
The percentage error in \( t \) is \( 11\% \).

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