If `L=20.04m pm 0.01m, B=2.52 m pm 0.02 m.` The value of `(LxxB)` is

Options

A. `(50.5pm 0.84)m^(2)`
B. `(50.5pm 0.42)m^(2)`
C. `(50.5pm0.21)m^(2)`
D. `(50.5pm0.50)m^(2)`

Explanation

To solve the problem of calculating the product of two measurements \( L \) and \( B \) along with their uncertainties, we will follow these steps:

### Step 1: Identify the values and uncertainties
We have:
- \( L = 20.04 \, \text{m} \pm 0.01 \, \text{m} \)
- \( B = 2.52 \, \text{m} \pm 0.02 \, \text{m} \)

### Step 2: Calculate the product \( X = L \times B \)
We need to calculate:
\[ X = L \times B = 20.04 \, \text{m} \times 2.52 \, \text{m} \]
Calculating this gives:
\[ X = 50.5088 \, \text{m}^2 \]
For practical purposes, we can round this to:
\[ X \approx 50.51 \, \text{m}^2 \]

### Step 3: Calculate the relative uncertainties
The formula for the propagation of uncertainty when multiplying two quantities is:
\[ \frac{\Delta X}{X} = \frac{\Delta L}{L} + \frac{\Delta B}{B} \]
Where:
- \( \Delta L = 0.01 \, \text{m} \)
- \( \Delta B = 0.02 \, \text{m} \)

Calculating the relative uncertainties:
\[ \frac{\Delta L}{L} = \frac{0.01}{20.04} \approx 0.000498 \]
\[ \frac{\Delta B}{B} = \frac{0.02}{2.52} \approx 0.007936 \]

### Step 4: Add the relative uncertainties
Now, we add these relative uncertainties:
\[ \frac{\Delta X}{X} \approx 0.000498 + 0.007936 \approx 0.008434 \]

### Step 5: Calculate the absolute uncertainty \( \Delta X \)
Now we can find \( \Delta X \):
\[ \Delta X = X \times \frac{\Delta X}{X} = 50.51 \, \text{m}^2 \times 0.008434 \approx 0.426 \, \text{m}^2 \]
We can round this to:
\[ \Delta X \approx 0.43 \, \text{m}^2 \]

### Step 6: Write the final result
Thus, the final value of \( L \times B \) with its uncertainty is:
\[ L \times B = 50.51 \, \text{m}^2 \pm 0.43 \, \text{m}^2 \]

### Final Answer:
\[ L \times B = 50.51 \pm 0.43 \, \text{m}^2 \]

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