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### Question 5: The Banquet Problem

You are tasked with arranging tables to seat 18 people. Each table seats 4, and you can place the tables in different configurations. Below is how we approach this:

#### Part (a): Different Possible Arrangements of Tables
The key here is that the tables can be arranged in either:
1. A long line: This would seat people along the sides, with 8 people per side and 1 person at each end.
2. Rectangular or other connected shapes: You can arrange them into a more compact form like a square or a U-shape.

##### Example Arrangements:
- **One long line of 8 tables**: Seats 18 (8 people on each side, plus 1 on each end).
- **Rectangular shape (2 by 4 tables)**: This arrangement would have 2 rows of 4 tables each. You could place people along the perimeter.
- **Square-like shape (2 by 2 or 3 by 3 clusters)**: Some of these arrangements may waste table edges or corners.

You would need to sketch these arrangements to find which uses the fewest tables.

#### Part (b): Least Expensive Arrangement
The arrangement that minimizes the number of tables would be the most cost-efficient. Based on the line formation, you need 8 tables. At $10 per table, this would cost **$80**.

If another arrangement used fewer tables (though unlikely given the problem's specifics), it would be cheaper.

### Question 6:
#### Part (a): Can two figures have the same area but different perimeters?
Yes, figures can have the same area but different perimeters. For example, consider:
- A **rectangle** of dimensions 6 by 4 (Area = 24, Perimeter = 20).
- A **square** of dimensions 5 by 5 (Area = 25, Perimeter = 20).

The two figures have different areas but similar perimeters.

#### Part (b): Can two figures have the same perimeter but different areas?
Yes, two figures can have the same perimeter but different areas. For example:
- A **rectangle** of dimensions 9 by 1 (Perimeter = 20, Area = 9).
- A **rectangle** of dimensions 5 by 5 (Perimeter = 20, Area = 25).

These figures have the same perimeter but different areas.

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Let me know if you need more details or sketches on these problems!

### Additional Questions:
1. What factors influence how the tables can be arranged for optimal seating?
2. Could alternative seating patterns (like U-shaped) help reduce the number of tables needed?
3. How does changing the shape of figures affect both the perimeter and area?
4. Why does a square often have minimal perimeter compared to other shapes with the same area?
5. Can you create an irregular shape with a larger perimeter but the same area as a rectangle?

### Tip:
For many area and perimeter problems, rectangles and squares give good benchmarks for comparison because of their regular properties.

Activity 2 (Relationships Between Perimeter and Area) Questions. Includes a banquet problem involving table arrangements and additional questions on the relationship between area and perimeter.

This activity explores the relationship between perimeter and area, focusing on a banquet seating arrangement problem. It covers table arrangements for minimizing cost and investigates how different shapes can have the same area but different perimeters.

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