If length and breath of a plate are `(40 +- 0.2)cm` and `(30 +- 0.1 ) cm`, the absolute error in the meaurement of area is

Options

A. `10 cm^(2)`
B. `8 cm^(2)`
C. `9cm^(2)`
D. `7cm^(2)`

Explanation

To find the absolute error in the measurement of the area of a plate given its length and breadth with their respective uncertainties, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Measurements and Their Uncertainties:**
- Length \( L = 40 \, \text{cm} \) with an uncertainty of \( \Delta L = 0.2 \, \text{cm} \)
- Breadth \( B = 30 \, \text{cm} \) with an uncertainty of \( \Delta B = 0.1 \, \text{cm} \)

2. **Calculate the Area:**
- The area \( A \) of the plate is given by the formula:
\[ A = L \times B \]
- Substituting the values:
\[ A = 40 \, \text{cm} \times 30 \, \text{cm} = 1200 \, \text{cm}^2 \]

3. **Determine the Relative Errors:**
- The relative error in length \( \left( \frac{\Delta L}{L} \right) \):
\[ \frac{\Delta L}{L} = \frac{0.2 \, \text{cm}}{40 \, \text{cm}} = 0.005 \]
- The relative error in breadth \( \left( \frac{\Delta B}{B} \right) \):
\[ \frac{\Delta B}{B} = \frac{0.1 \, \text{cm}}{30 \, \text{cm}} \approx 0.00333 \]

4. **Calculate the Relative Error in Area:**
- The relative error in area \( \left( \frac{\Delta A}{A} \right) \) can be calculated using the formula:
\[ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} \]
- Substituting the values:
\[ \frac{\Delta A}{A} = 0.005 + 0.00333 \approx 0.00833 \]

5. **Calculate the Absolute Error in Area:**
- The absolute error \( \Delta A \) can be calculated using:
\[ \Delta A = A \times \frac{\Delta A}{A} \]
- Substituting the values:
\[ \Delta A = 1200 \, \text{cm}^2 \times 0.00833 \approx 10 \, \text{cm}^2 \]

### Final Result:
The absolute error in the measurement of the area is \( \Delta A \approx 10 \, \text{cm}^2 \).

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