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This document outlines a quadratic optimization project involving the construction of sheep pens with the goal of maximizing the pen area using a limited amount of fencing. Below is a summary of key points and mathematical tasks required for the investigation:

### Background:
- **Objective**: Find the optimal pen designs that maximize the area to hold as many sheep as possible within a fixed amount of fencing.
- **Optimization method**: Using quadratic equations to solve the problem.

### Task Overview:
You are asked to:
1. **Choose the amount of fencing** available, ranging from 150m to 350m.
2. **Research land requirements per sheep** to calculate the number of sheep that can be housed.

### Problem Options:
#### **Part A:**
- **Option 1**: Design a rectangular pen and find the shape that maximizes the area. The mathematical basis for the optimal solution needs explanation.
- **Option 2**: A brick wall, 200m long, can be used as one side of the pen, reducing the required fencing. You need to find the design that maximizes the area.
- **Option 3**: Consider using natural obstacles such as a river or rocks as part of the pen boundary. This option allows you to design a solution with reduced fencing.

#### **Part B**: 
- The pen must be divided into two compartments (for ewes and rams), possibly requiring different space allocations. You need to find optimal designs for the split in both Option 1 and Option 2.

#### **Part C**: 
- Explore splitting the pens into multiple compartments (3, 4, 5, etc.). Investigate patterns that emerge in terms of maximizing the total area for each design. Analyze how changes to the length of the brick wall (e.g., if it's only 30m) impact the results.

### Final Report:
The project concludes with a report explaining the methods, calculations, and critical analysis of each design's results. You must provide clear reasoning, results, and a discussion of limitations or assumptions.

Would you like me to help solve any specific task, or should we focus on one of the options? Feel free to ask for clarifications.

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Here are five relative questions for deeper exploration:
1. How do you derive the optimal area for the rectangular pen in Option 1?
2. What impact does using the 200m brick wall have on the area in Option 2?
3. How would adding natural obstacles influence the design and sheep capacity?
4. How can quadratic equations be applied to the problem of splitting the pen into multiple compartments?
5. What assumptions might limit the applicability of the optimal solutions?

**Tip**: For quadratic optimization, always start by writing the equation for the area in terms of the fencing length to simplify the analysis.

This document outlines a quadratic optimization project involving the construction of sheep pens with the goal of maximizing the pen area using a limited amount of fencing.

Explore how to maximize the area of sheep pens using quadratic equations with limited fencing. Includes different design options such as rectangular pens, brick wall integration, and natural obstacle consideration. Suitable for Grades 10-12.

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