If length and breadth of a plane are `(40 + 0.2)` and `(30 +- 0.1)` cm, the absolute error in measurement of area is

Options

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

Answer

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

### Step 1: Identify the given values
- Length (L) = \(40 \pm 0.2\) cm
- Breadth (B) = \(30 \pm 0.1\) cm

### Step 2: Calculate the actual area
The area (A) of a rectangle is given by the formula:
\[ A = L \times B \]
Substituting the values:
\[ A = 40 \, \text{cm} \times 30 \, \text{cm} = 1200 \, \text{cm}^2 \]

### Step 3: Determine the relative errors
The relative error in the area can be calculated using the formula:
\[ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} \]
Where:
- \(\Delta L = 0.2\) cm (absolute error in length)
- \(\Delta B = 0.1\) cm (absolute error in breadth)

### Step 4: Substitute the values into the formula
Now we can substitute the values into the relative error formula:
\[ \frac{\Delta A}{1200} = \frac{0.2}{40} + \frac{0.1}{30} \]

### Step 5: Calculate each term
Calculating the first term:
\[ \frac{0.2}{40} = 0.005 \]
Calculating the second term:
\[ \frac{0.1}{30} \approx 0.00333 \]

### Step 6: Add the relative errors
Now, we add these two relative errors:
\[ \frac{\Delta A}{1200} = 0.005 + 0.00333 \approx 0.00833 \]

### Step 7: Calculate the absolute error in area
To find the absolute error in area (\(\Delta A\)), we multiply the relative error by the actual area:
\[ \Delta A = A \times \frac{\Delta A}{A} = 1200 \times 0.00833 \approx 10 \, \text{cm}^2 \]

### Conclusion
The absolute error in the measurement of the area is approximately \(10 \, \text{cm}^2\).

---