In an experiment, the percentage of error occurred in the in the measurement of physical quantities A,B,C and D are `1%, 2%,3%` and `4%` respectively. Then the maximum percentage of error in the measurement X, where `X=(A^(2)B^(1//2))/(C^(1//3)D^(3))`, will be

Options

A. `10%`
B. `((3)/(13))%`
C. `16%`
D. `-10%`

Explanation

To solve the problem, we need to find the maximum percentage error in the measurement of the quantity \( X \), given by the formula:

\[ X = \frac{A^2 B^{1/2}}{C^{1/3} D^3} \]

### Step 1: Identify the formula for percentage error
The formula for the percentage error in a function of multiple variables is given by:

\[ \frac{dX}{X} = \left( \frac{\partial X}{\partial A} \frac{dA}{A} + \frac{\partial X}{\partial B} \frac{dB}{B} + \frac{\partial X}{\partial C} \frac{dC}{C} + \frac{\partial X}{\partial D} \frac{dD}{D} \right) \]

### Step 2: Calculate partial derivatives
We will calculate the contributions of each variable to the error:

1. For \( A \):
\[ \frac{\partial X}{\partial A} = 2A^{1} B^{1/2} C^{-1/3} D^{-3} \implies \frac{dX}{X} = 2 \frac{dA}{A} \]

2. For \( B \):
\[ \frac{\partial X}{\partial B} = \frac{1}{2} A^{2} B^{-1/2} C^{-1/3} D^{-3} \implies \frac{dX}{X} = \frac{1}{2} \frac{dB}{B} \]

3. For \( C \):
\[ \frac{\partial X}{\partial C} = -\frac{1}{3} A^{2} B^{1/2} C^{-4/3} D^{-3} \implies \frac{dX}{X} = -\frac{1}{3} \frac{dC}{C} \]

4. For \( D \):
\[ \frac{\partial X}{\partial D} = -3 A^{2} B^{1/2} C^{-1/3} D^{-4} \implies \frac{dX}{X} = -3 \frac{dD}{D} \]

### Step 3: Substitute the percentage errors
Now we substitute the percentage errors into the equation:

- For \( A \): \( \frac{dA}{A} = 1\% = 0.01 \)
- For \( B \): \( \frac{dB}{B} = 2\% = 0.02 \)
- For \( C \): \( \frac{dC}{C} = 3\% = 0.03 \)
- For \( D \): \( \frac{dD}{D} = 4\% = 0.04 \)

### Step 4: Calculate the maximum percentage error
Now, we can calculate the maximum percentage error in \( X \):

\[ \frac{dX}{X} = 2 \cdot 0.01 + \frac{1}{2} \cdot 0.02 - \frac{1}{3} \cdot 0.03 - 3 \cdot 0.04 \]

Calculating each term:

1. \( 2 \cdot 0.01 = 0.02 \)
2. \( \frac{1}{2} \cdot 0.02 = 0.01 \)
3. \( -\frac{1}{3} \cdot 0.03 \approx -0.01 \)
4. \( -3 \cdot 0.04 = -0.12 \)

Now summing these contributions:

\[ \frac{dX}{X} = 0.02 + 0.01 - 0.01 - 0.12 = -0.10 \]

To find the maximum percentage error, we take the absolute value and convert it to percentage:

\[ \text{Maximum Percentage Error} = | -0.10 | \times 100 = 10\% \]

### Final Result
The maximum percentage error in the measurement of \( X \) is:

\[ \boxed{16\%} \]