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The problem describes a rectangular plate with the following dimensions:
- Length: \( (21.2 \pm 0.2) \, \text{cm} \)
- Width: \( (10.8 \pm 0.1) \, \text{cm} \)

We are tasked with calculating the area of the plate and its uncertainty.

### Step 1: Calculate the Area of the Rectangle
The formula for the area \( A \) of a rectangle is:

\[
A = \text{Length} \times \text{Width}
\]

Using the given values:

\[
A = 21.2 \, \text{cm} \times 10.8 \, \text{cm}
\]

Let's calculate this area first.

### Step 2: Calculate the Uncertainty of the Area
The uncertainty in the area \( \Delta A \) for multiplication of two quantities is given by:

\[
\frac{\Delta A}{A} = \sqrt{\left(\frac{\Delta L}{L}\right)^2 + \left(\frac{\Delta W}{W}\right)^2}
\]

Where:
- \( \Delta L = 0.2 \, \text{cm} \) (uncertainty in length)
- \( \Delta W = 0.1 \, \text{cm} \) (uncertainty in width)
- \( L = 21.2 \, \text{cm} \) (length)
- \( W = 10.8 \, \text{cm} \) (width)

Let's calculate the area and the uncertainty.import math

# Given values
L = 21.2  # length in cm
W = 10.8  # width in cm
delta_L = 0.2  # uncertainty in length in cm
delta_W = 0.1  # uncertainty in width in cm

# Step 1: Calculate the area
A = L * W  # area of the rectangle

# Step 2: Calculate the uncertainty in the area
relative_uncertainty = math.sqrt((delta_L / L) ** 2 + (delta_W / W) ** 2)
delta_A = relative_uncertainty * A  # uncertainty in area

A, delta_AThe area of the rectangular plate is approximately \( 228.96 \, \text{cm}^2 \).

The uncertainty in the area is approximately \( \pm 3.03 \, \text{cm}^2 \).

Thus, the area of the plate, including its uncertainty, is:
\[
A = (228.96 \pm 3.03) \, \text{cm}^2
\]

Do you want more details on any of the steps or have any questions?

Here are some related questions for further exploration:
1. How would the area change if the length increased by 5%?
2. What is the perimeter of the rectangular plate with its uncertainty?
3. How does the uncertainty in width impact the overall uncertainty in the area?
4. If the width is measured with higher precision, what happens to the uncertainty in the area?
5. How can we minimize uncertainties in measurements like this?

**Tip:** When working with uncertainties, always track the relative uncertainties in your measurements, as this gives a clearer picture of how much the final result might vary.

Learn how to calculate the area of a rectangular plate with uncertainty. Understand the method to determine the area and its uncertainty using measurements. Includes step-by-step calculations and examples. Suitable for advanced high school students and anyone interested in precision measurement and error analysis.

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