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!