The Young modulus (Y) of a material is given by `Y=(WL)/(pir^(2)l)`. If the percentage error in W, L, r and l are `0.5%, 1%, 3%" and "4%` respectively then maximum percentage error in Y is -

Options

A. `7.5%`
B. `9%`
C. `11.5%`
D. `13%`

Answer

To find the maximum percentage error in Young's modulus (Y), we can use the formula for the percentage error in a function of multiple variables. The formula for Young's modulus is given as:

\[ Y = \frac{WL}{\pi r^2 l} \]

where:
- \( W \) is the force applied,
- \( L \) is the length,
- \( r \) is the radius,
- \( l \) is the original length of the material.

### Step 1: Identify the percentage errors
We are given the percentage errors in the measurements:
- Percentage error in \( W \) = \( 0.5\% \)
- Percentage error in \( L \) = \( 1\% \)
- Percentage error in \( r \) = \( 3\% \)
- Percentage error in \( l \) = \( 4\% \)

### Step 2: Write the formula for maximum percentage error in Y
The percentage error in Y can be calculated using the following formula:

\[ \text{Percentage error in } Y = \left( \frac{\Delta W}{W} + \frac{\Delta L}{L} + 2 \frac{\Delta r}{r} + \frac{\Delta l}{l} \right) \times 100 \]

### Step 3: Substitute the given percentage errors
Substituting the given values into the formula:

\[ \text{Percentage error in } Y = 0.5\% + 1\% + 2 \times 3\% + 4\% \]

### Step 4: Calculate the total percentage error
Now, calculate the total:

\[ \text{Percentage error in } Y = 0.5 + 1 + 6 + 4 \]
\[ = 0.5 + 1 + 6 + 4 = 11.5\% \]

### Conclusion
Thus, the maximum percentage error in Young's modulus (Y) is:

\[ \boxed{11.5\%} \]