the length and breadth of a rectangular metal sheet measured using meter ruler marked in mm are 20.0cm and 12.5cm respectively. calculate the possible error of the area and the percentage possible error of area

### Step 1: Understand the measurements and potential error.
- Length (\( L \)): \( 20.0 \, \text{cm} \)
- Breadth (\( B \)): \( 12.5 \, \text{cm} \)
- The measurements are made using a meter ruler marked in millimeters (mm), meaning the smallest unit of measurement is \( 1 \, \text{mm} \) or \( 0.1 \, \text{cm} \).

The possible error in each measurement is \( \pm 0.1 \, \text{cm} \).

### Step 2: Calculate the area.
The area \( A \) of a rectangle is given by:
\[
A = L \times B
\]
Substitute the given values:
\[
A = 20.0 \, \text{cm} \times 12.5 \, \text{cm} = 250.0 \, \text{cm}^2
\]

### Step 3: Calculate the possible error in the area.
The possible error in the area (\( \Delta A \)) can be estimated using the formula:
\[
\frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B}
\]
Rearranging to find \( \Delta A \):
\[
\Delta A = A \times \left( \frac{\Delta L}{L} + \frac{\Delta B}{B} \right)
\]
Substitute the given values:
\[
\frac{\Delta L}{L} = \frac{0.1 \, \text{cm}}{20.0 \, \text{cm}} = 0.005
\]
\[
\frac{\Delta B}{B} = \frac{0.1 \, \text{cm}}{12.5 \, \text{cm}} = 0.008
\]
\[
\Delta A = 250.0 \, \text{cm}^2 \times (0.005 + 0.008) = 250.0 \, \text{cm}^2 \times 0.013 = 3.25 \, \text{cm}^2
\]

### Step 4: Calculate the percentage error.
The percentage error in the area is given by:
\[
\text{Percentage Error} = \left( \frac{\Delta A}{A} \right) \times 100\%
\]
Substitute the values:
\[
\text{Percentage Error} = \left( \frac{3.25 \, \text{cm}^2}{250.0 \, \text{cm}^2} \right) \times 100\% = 1.3\%
\]

### Final Answers:
- **Possible Error in Area:** \( \pm 3.25 \, \text{cm}^2 \)
- **Percentage Error in Area:** \( 1.3\% \)

Would you like further details or have any questions?

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Here are 5 related questions to explore:

1. How would the possible error in the area change if the length was 25.0 cm?
2. What is the possible error in the perimeter of the metal sheet?
3. How does the percentage error in the area compare to the percentage error in the perimeter?
4. What happens to the possible error in the area if the ruler's smallest unit of measurement is changed to 0.05 cm?
5. How would the errors change if the measurements were made in inches instead of centimeters?

**Tip:** When calculating errors for derived quantities like area or volume, remember that the relative errors of the measurements are additive in such calculations.

Learn how to calculate the possible error and percentage error of the area of a rectangle when measurements are made using a meter ruler marked in millimeters. Example problem: Length = 20.0 cm, Breadth = 12.5 cm. Possible error in area: ±3.25 cm², Percentage error in area: 1.3%.

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