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 and its associated uncertainty, we can follow these steps:

### Step 1: Identify the given values
- Length (L) = \(5 \pm 0.1\) cm
- Breadth (b) = \(2 \pm 0.01\) cm

### Step 2: Calculate the area
The area (A) of a rectangle is given by the formula:
\[ A = L \times b \]
Substituting the values:
\[ A = 5 \, \text{cm} \times 2 \, \text{cm} = 10 \, \text{cm}^2 \]

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

### Step 4: Substitute the values into the uncertainty formula
Substituting the known values:
\[ \frac{\Delta A}{10} = \frac{0.1}{5} + \frac{0.01}{2} \]

Calculating each term:
- \(\frac{0.1}{5} = 0.02\)
- \(\frac{0.01}{2} = 0.005\)

Adding these together:
\[ \frac{\Delta A}{10} = 0.02 + 0.005 = 0.025 \]

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

### Step 6: Write the final result
The area of the plate with its uncertainty is:
\[ A \pm \Delta A = 10 \pm 0.25 \, \text{cm}^2 \]

### Final Answer:
The area of the plate is \(10 \pm 0.25 \, \text{cm}^2\).

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