What is the largest rectangular chicken pen (enclosure) that a farmer can construct (fence) if he/she is providing with a wire fencing of 20 metres? The farmer is expected to use only full metres for the sides.

1. Use Polya to solve problem (8)
2. Use three representations(picture/diagrams, table and graph) to model the problem (6()
3. What assumption did you make to model situation (2)
4. If the farmer fenced circular chicken pen using 20 metres of metres of wire fencing, what would the largest area of the chicken pen be? (3)

not sure about Polya, but the maximum-area quadrilateral is a square.

as for the circle, since
c = 2πr and A = πr^2,
A = c^2/(4π)

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Step 1: Use Polya to solve the problem (8)

1. Understand the problem: The farmer has been provided with 20 meters of wire fencing and needs to construct the largest rectangular chicken pen using only full meters for the sides.

2. Devise a plan: To find the largest rectangular chicken pen, we need to determine the maximum area. We will start by considering different possible dimensions and calculating the area for each scenario.

3. Carry out the plan: We will explore possible side lengths for the rectangular pen and calculate the area using various dimensions. By comparing the areas, we can determine the largest possible area.

4. Look back: After finding the maximum area, we will answer the question by providing a final solution.

Step 2: Use three representations (picture/diagrams, table, and graph) to model the problem (6)

1. Picture/Diagram representation: Draw rectangles of different dimensions to visualize the chicken pens. Label the length and width for each scenario and calculate the area.

2. Table representation:

| Length (m) | Width (m) | Area (m²) |
|------------|-----------|-----------|
| 1 | 9 | 9 |
| 2 | 8 | 16 |
| 3 | 7 | 21 |
| 4 | 6 | 24 |
| 5 | 5 | 25 |
| 6 | 4 | 24 |
| 7 | 3 | 21 |
| 8 | 2 | 16 |
| 9 | 1 | 9 |

3. Graph representation: Plot the area against the length of the rectangular pen on a graph.

Step 3: Assumption made to model the situation (2)

The assumption made to model the situation is that the rectangular chicken pen must have sides with full meters. In other words, the length and width of the pen can only be whole numbers.

Step 4: If the farmer fenced a circular chicken pen using 20 meters of wire fencing, what would the largest area of the chicken pen be? (3)

Since a circular pen has the largest area among all shapes with the same perimeter, we can calculate the area using the formula: A = πr², where r is the radius of the circular pen. In this case, the perimeter is given as 20 meters.

Perimeter of a circle = 2πr
20 = 2πr
r = 20 / (2π)
r ≈ 3.183

The largest area of the circular chicken pen would be A ≈ π(3.183)² ≈ 31.779 square meters.

To answer these questions, we will follow the steps provided.

1. Use Polya to solve the problem:
a. Understand the problem: The problem is asking for the largest rectangular chicken pen (enclosure) that a farmer can construct with a wire fencing of 20 metres, using only full meters for the sides.
b. Devise a plan: To find the largest rectangular chicken pen, we need to determine the dimensions of the rectangle that maximize its area.
c. Carry out the plan: We can try different dimensions and calculate the area to find the maximum.
d. Look back: Ensure the solution aligns with the problem statement.

To calculate the largest rectangular chicken pen, we need to consider that the wire fencing is only available in full meters. This means that the length and width of the pen must be whole numbers.

We can start by trying different values for the length and width of the rectangular pen and checking their areas:

Length = 1 meter, Width = 9 meters ⇒ Area = 1 * 9 = 9 square meters
Length = 2 meters, Width = 8 meters ⇒ Area = 2 * 8 = 16 square meters
Length = 3 meters, Width = 7 meters ⇒ Area = 3 * 7 = 21 square meters
Length = 4 meters, Width = 6 meters ⇒ Area = 4 * 6 = 24 square meters

Continuing with this process, we can find the largest rectangular chicken pen.

2. Use three representations (picture/diagrams, table, and graph) to model the problem:
a. Picture/Diagram: Draw a rectangle representing the chicken pen with labeled dimensions.
b. Table: Create a table with the dimensions, areas, and corresponding values for length and width.
c. Graph: Plot the dimensions on a graph with the length on the x-axis and the area on the y-axis.

3. Assumptions made to model the situation:
We assume that the farmer can construct a rectangular chicken pen using any combination of whole numbers for the length and width, as long as the total length of wire fencing is 20 meters.

4. If the farmer fenced a circular chicken pen using 20 meters of wire fencing, what would the largest area of the chicken pen be?
To solve this, we need to use the formula for the area of a circle:

Area = π * r^2

In this case, the total length of wire fencing is given as 20 meters. So, we can use the formula for the circumference of a circle to find the radius:

Circumference = 2 * π * r

Given that the circumference is 20 meters, we can solve for r:

20 = 2 * π * r
r = 20 / (2 * π)

Now, substitute the radius value into the area formula:

Area = π * (20 / (2 * π))^2

Simplifying the expression:

Area = (400 / (4 * π))
Area = 100 / π

So, the largest area of the circular chicken pen with 20 meters of wire fencing would be 100 / π square meters.