A plate has a length `( 5pm 0.1 )` cm and breadth `(2pm 0.01)` cm . Then the area of the plate is

Options

A. `(10pm0.25)cm^2`
B. `(10pm0.01)cm^2`
C. `(10pm0.001)cm^2`
D. `(10pm1)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 Length and Breadth
The length \( L \) of the plate is given as:
\[ L = 5 \pm 0.1 \text{ cm} \]
The breadth \( B \) of the plate is given as:
\[ B = 2 \pm 0.01 \text{ cm} \]

### Step 2: Calculate the Area
The area \( A \) of the plate can be calculated using the formula:
\[ A = L \times B \]
Substituting the values:
\[ A = 5 \times 2 = 10 \text{ cm}^2 \]

### Step 3: Calculate the Uncertainty in Area
To find the uncertainty in the area \( \Delta A \), we use the formula for the propagation of uncertainty in multiplication:
\[ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} \]
Where:
- \( \Delta L = 0.1 \text{ cm} \)
- \( \Delta B = 0.01 \text{ cm} \)

Substituting the values:
\[ \frac{\Delta A}{10} = \frac{0.1}{5} + \frac{0.01}{2} \]

### Step 4: Simplify the Expression
Calculating each term:
\[ \frac{0.1}{5} = 0.02 \]
\[ \frac{0.01}{2} = 0.005 \]
Now, adding these:
\[ \frac{\Delta A}{10} = 0.02 + 0.005 = 0.025 \]

### Step 5: Solve for \( \Delta A \)
Now, multiply both sides by 10 to find \( \Delta A \):
\[ \Delta A = 10 \times 0.025 = 0.25 \text{ cm}^2 \]

### Step 6: Final Area with Uncertainty
Thus, the area of the plate with its uncertainty is:
\[ A = 10 \pm 0.25 \text{ cm}^2 \]

### Conclusion
The final answer is:
\[ \text{Area} = 10 \pm 0.25 \text{ cm}^2 \]