A physical quantity `X `is give by the relation `X = (2h^(3)I^(2))/(2sqrt(n))` The percentage error in the meansurement of k ,I,m and n are `1% ,2%, 3%` and `4%` respectively The value of X is uncertain by

Options

A. 0.08
B. 0.1
C. 0.12
D. None of these

Explanation

To solve the problem, we need to calculate the percentage uncertainty in the physical quantity \( X \) given by the formula:

\[ X = \frac{2h^3 I^2}{2\sqrt{n}} \]

However, we can simplify the expression to:

\[ X = \frac{h^3 I^2}{\sqrt{n}} \]

Next, we need to find the percentage error in \( X \) based on the percentage errors in the measurements of \( h \), \( I \), and \( n \). The percentage errors are given as follows:

- Percentage error in \( h \) (denoted as \( \delta h \)): 1%
- Percentage error in \( I \) (denoted as \( \delta I \)): 2%
- Percentage error in \( n \) (denoted as \( \delta n \)): 4%

### Step 1: Write the formula for percentage error in \( X \)

The formula for the percentage error in a product or quotient of quantities is given by:

\[ \text{Percentage error in } X = \left( \text{sum of the percentage errors of the variables involved} \right) \]

For our equation \( X = \frac{h^3 I^2}{\sqrt{n}} \), we can express the percentage error in \( X \) as:

\[ \text{Percentage error in } X = 3 \cdot \text{Percentage error in } h + 2 \cdot \text{Percentage error in } I + \frac{1}{2} \cdot \text{Percentage error in } n \]

### Step 2: Substitute the values of percentage errors

Now, substituting the given percentage errors:

- For \( h \): \( 1\% \)
- For \( I \): \( 2\% \)
- For \( n \): \( 4\% \)

We can write:

\[ \text{Percentage error in } X = 3 \cdot 1\% + 2 \cdot 2\% + \frac{1}{2} \cdot 4\% \]

### Step 3: Calculate each term

Calculating each term:

1. \( 3 \cdot 1\% = 3\% \)
2. \( 2 \cdot 2\% = 4\% \)
3. \( \frac{1}{2} \cdot 4\% = 2\% \)

### Step 4: Sum the contributions

Now, adding these contributions together:

\[ \text{Percentage error in } X = 3\% + 4\% + 2\% = 9\% \]

### Final Answer

Thus, the value of \( X \) is uncertain by **9%**.

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