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

Options

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

Explanation

To find the percentage error in \( t \) given the formula \( t = \frac{xy^2}{z^2} \) and the percentage errors in \( x \), \( y \), and \( z \), we can follow these steps:

### Step 1: Identify the formula for \( t \)
The formula given is:
\[ t = \frac{xy^2}{z^2} \]

### Step 2: Write down the percentage errors
We are given the following percentage errors:
- Percentage error in \( x \) (\( \delta x \)) = 1%
- Percentage error in \( y \) (\( \delta y \)) = 3%
- Percentage error in \( z \) (\( \delta z \)) = 2%

### Step 3: Use the formula for percentage error in \( t \)
The formula for the percentage error in a function of multiple variables is given by:
\[ \frac{\delta t}{t} \times 100 = \frac{\delta x}{x} \times 100 + n \cdot \frac{\delta y}{y} \times 100 - m \cdot \frac{\delta z}{z} \times 100 \]
where \( n \) and \( m \) are the powers of \( y \) and \( z \) in the formula for \( t \).

### Step 4: Identify the powers
In our case:
- The power of \( y \) is 2 (since \( y^2 \)).
- The power of \( z \) is 2 (since \( z^2 \)).

### Step 5: Substitute the values into the formula
Now we substitute the values into the formula:
\[ \frac{\delta t}{t} \times 100 = \frac{\delta x}{x} \times 100 + 2 \cdot \frac{\delta y}{y} \times 100 + 2 \cdot \frac{\delta z}{z} \times 100 \]
Substituting the given percentage errors:
\[ \frac{\delta t}{t} \times 100 = 1 + 2 \cdot 3 + 2 \cdot 2 \]

### Step 6: Calculate the total percentage error
Calculating the right-hand side:
\[ \frac{\delta t}{t} \times 100 = 1 + 6 + 4 = 11 \]

### Step 7: Conclusion
Thus, the percentage error in \( t \) is:
\[ \delta t = 11\% \]

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

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