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It seems that you've uploaded an image with a mathematical problem about optimizing the area of a rectangular pen for a pig, given a fixed perimeter of 20 meters. Let's break down the problem and solve it.

### Part 1: Completing the chart
The task involves filling out a chart where:
- **Width of the fence (W)** varies from 1 to 9 meters.
- The **length of the fence (L)** is calculated from the total perimeter formula:  
  $ 2L + 2W = 20 \implies L = 10 - W $.
- The **enclosed area** is given by the formula:  
  $ \text{Area} = L \times W = (10 - W) \times W = 10W - W^2 $.

Let me now complete the missing row from the chart.

- For **Width = 8 meters**:
  - Length of Fence = $ 10 - 8 = 2 $ meters.
  - Enclosed Area = $ 8 \times 2 = 16 $ square meters.

The missing entry is 16 square meters for the enclosed area when the width is 8 meters.

### Part 2: Solving the questions
#### a) What dimensions produce the largest area?
To find the maximum area, we can observe the table. The area is maximum when the width is 5 meters, and the length is 5 meters, yielding an area of 25 square meters. So, a square pen with sides 5 meters long produces the largest area.

#### b) What dimensions produce the smallest area?
The smallest area occurs when the width is either 1 meter or 9 meters, with corresponding areas of 9 square meters.

#### c) Why are all your answers in column two subtracted from 20 in column 3?
The values in column 3 represent the length of the fence, calculated using the formula $ L = 10 - W $, where 20 meters is the total perimeter of the fence. Since the perimeter is split between the length and width, the length is calculated by subtracting the total width values from 20 (half of the perimeter).

#### d) Why is the largest value in the first column only 9 meters?
The largest value for the width is 9 meters because the total perimeter is 20 meters. Since both the width and length are part of the perimeter, when the width is 9 meters, the corresponding length is 1 meter. Beyond this, the dimensions wouldn't satisfy the perimeter constraint.

#### e) Fill in the blank: "The shape of the pen that produces the largest area is a **square**."

This is because a square (equal length and width) maximizes the area for a given perimeter, as shown by the calculations.

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Would you like further details on any of the questions, or need help with another part?

### 5 Related Questions:
1. How would the area change if the total perimeter was increased to 30 meters?
2. Can you derive the formula for the area in terms of the width from the perimeter equation?
3. What is the geometric reasoning behind why a square maximizes the area?
4. How does the maximum area change if the pen is not rectangular but circular?
5. If the fencing cost is $5 per meter, how much would the fence cost for the optimal dimensions?

### Tip:
When optimizing areas under perimeter constraints, it often helps to consider symmetry (as in the case of squares or circles), as symmetric shapes tend to maximize area for a given boundary.

A farmer has 20 meters of fencing to create an enclosed rectangular pen for his pig. The goal is to determine the dimensions that maximize the area of the pen.

Solve a real-world optimization problem: determine the dimensions of a rectangular pen that yield the largest enclosed area, given a fixed perimeter of 20 meters. Explore algebraic concepts such as quadratic equations and area maximization.

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