https://file.aiquickdraw.com/mathbot/file/1727473390996up9v52y4.jpg

To solve this problem, we need to determine the lower and upper bounds of the area of region \( R \). The image gives two key pieces of information:

- The **length** of region \( R \) is \( 3.1 \, \text{m} \) to the nearest \( 0.1 \, \text{m} \).
- The **area** of region \( R \) is \( 490 \, \text{cm}^2 \) to the nearest \( 10 \, \text{cm}^2 \).

### Step 1: Determine the bounds of the length
The length is given as \( 3.1 \, \text{m} \) to the nearest \( 0.1 \, \text{m} \), which means:
- The **upper bound** of the length is \( 3.1 + 0.05 = 3.15 \, \text{m} \).
- The **lower bound** of the length is \( 3.1 - 0.05 = 3.05 \, \text{m} \).

### Step 2: Determine the bounds of the area
The area is given as \( 490 \, \text{cm}^2 \) to the nearest \( 10 \, \text{cm}^2 \), which means:
- The **upper bound** of the area is \( 490 + 5 = 495 \, \text{cm}^2 \).
- The **lower bound** of the area is \( 490 - 5 = 485 \, \text{cm}^2 \).

### Step 3: Convert the area from cm² to m²
Since the length is in meters and the area is in square centimeters, we need to convert the area of \( R \) into square meters:
\[
1 \, \text{m}^2 = 10,000 \, \text{cm}^2
\]
So, converting the bounds of the area to square meters:
- The **lower bound** of the area in m² is:
\[
485 \, \text{cm}^2 = \frac{485}{10,000} = 0.0485 \, \text{m}^2
\]
- The **upper bound** of the area in m² is:
\[
495 \, \text{cm}^2 = \frac{495}{10,000} = 0.0495 \, \text{m}^2
\]

Thus, the lower and upper bounds for the area of \( R \) are approximately:

- **Lower bound**: \( 0.0485 \, \text{m}^2 \)
- **Upper bound**: \( 0.0495 \, \text{m}^2 \)

Let me know if you'd like more details or have further questions!

Here are 5 related questions you could explore:
1. How do rounding errors affect area calculations?
2. How can we calculate the perimeter of region \( R \)?
3. How would the solution change if different units were used?
4. How is the concept of significant figures related to bounds?
5. Can we estimate the volume of a 3D shape using similar bounds?

**Tip**: Always ensure units are consistent when calculating areas or volumes!

What are the lower and upper bounds of the area of R? Give your answers to the nearest 1 cm².

Learn how to calculate the lower and upper bounds of the area of a shape using rounding and unit conversions. This example demonstrates rounding length to the nearest 0.1 m and area to the nearest 10 cm², suitable for Grades 9-11 geometry students.

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