The length and breadth of a plate are `(6 pm 0.1) cm and (4 pm 0.2)cm` respectively. The area of the plate is

Options

A. `(24 pm 1.6)cm^2`
B. `24.4 cm^2`
C. `23.6 cm^2`
D. `(24 + 0.02) cm^2`

Answer

To find the area of the plate given its length and breadth along with their uncertainties, we can follow these steps:

### Step 1: Identify the given values
The length (L) of the plate is given as:
\[ L = 6 \pm 0.1 \, \text{cm} \]
The breadth (B) of the plate is given as:
\[ B = 4 \pm 0.2 \, \text{cm} \]

### Step 2: Calculate the area of the plate
The area (A) of the plate can be calculated using the formula:
\[ A = L \times B \]
Substituting the values:
\[ A = 6 \, \text{cm} \times 4 \, \text{cm} = 24 \, \text{cm}^2 \]

### Step 3: Calculate the uncertainties in the area
To find the uncertainty in the area (ΔA), we need to consider the percentage errors in the length and breadth.

1. **Calculate the percentage error in length (ΔL):**
\[ \text{Percentage error in } L = \frac{\Delta L}{L} \times 100 = \frac{0.1}{6} \times 100 \approx 1.67\% \]

2. **Calculate the percentage error in breadth (ΔB):**
\[ \text{Percentage error in } B = \frac{\Delta B}{B} \times 100 = \frac{0.2}{4} \times 100 = 5\% \]

3. **Total percentage error in area:**
The area is a product of length and breadth, so the total percentage error in area is the sum of the percentage errors in length and breadth:
\[ \text{Percentage error in } A = \text{Percentage error in } L + \text{Percentage error in } B = 1.67\% + 5\% = 6.67\% \]

### Step 4: Calculate the absolute uncertainty in area (ΔA)
Now we can calculate the absolute uncertainty in the area using the percentage error:
\[ \Delta A = \left( \frac{\text{Percentage error in } A}{100} \right) \times A = \left( \frac{6.67}{100} \right) \times 24 \approx 1.6 \, \text{cm}^2 \]

### Step 5: Write the final result
Thus, the area of the plate with its uncertainty is:
\[ A = 24 \pm 1.6 \, \text{cm}^2 \]

### Conclusion
The area of the plate is \( 24 \pm 1.6 \, \text{cm}^2 \).

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