Rick has 1200 feet of fence that he wants to use to enclose a rectangular garden. Find the maximum area the garden can have.

I suspect a square

A = L w
L + w = constant = k

L = k - w
so
A = (k-w)w = k w - w^2
dA/dw = k - 2w
= 0 for max or min
so
w = k/2
or w = k = k/2 sure enough a square
so
1200/4 = 300 = L = w
A = 90,000

To find the maximum area, we need to determine the dimensions of the rectangular garden that will use up all the available fence.

Let's assume the length of the garden is x feet.
Since the garden is enclosed with a fence, this means we need two sides of length x and two sides of length (1200 - 2x) to make up the rectangular shape.

The perimeter of a rectangle, which is the sum of all its sides, is given by 2(length + width).
In this case, the perimeter is equal to the amount of fence available, which is 1200 feet.
So we have the equation: 2(x + (1200 - 2x)) = 1200.

Let's solve this equation step-by-step:

1. Distribute the 2 on the left side of the equation:
2x + 2400 - 4x = 1200

2. Combine like terms:
-2x + 2400 = 1200

3. Move 2400 to the right side of the equation by subtracting it from both sides:
-2x = 1200 - 2400

4. Simplify:
-2x = -1200

5. Divide both sides of the equation by -2 to solve for x:
x = -1200 / -2
x = 600

We have found that the length of the garden should be 600 feet.

To find the width of the garden, we can substitute the value of x into (1200 - 2x):
width = 1200 - 2(600)
width = 1200 - 1200
width = 0

The width of the garden is 0 feet.

However, a garden with zero width doesn't make sense, so we need to recheck our calculations.

Let's analyze the equation 2(x + (1200 - 2x)) = 1200:

Using the distributive property, we can simplify this equation to:
2x + 2400 - 4x = 1200

Combine like terms:
-2x + 2400 = 1200

Subtract 2400 from both sides:
-2x = 1200 - 2400
-2x = -1200

Divide both sides by -2:
x = -1200 / -2
x = 600

It seems we made an error in our calculations. Let's redo the calculations correctly.

The length of the garden is x = 600 feet.
To find the width, we substitute this value into (1200 - 2x):
width = 1200 - 2(600)
width = 1200 - 1200
width = 0

The width turns out to be 0, which means the garden cannot have a width. Therefore, it is not possible to enclose a rectangular garden with the given fence length of 1200 feet.

To find the maximum area of the garden, we need to maximize the dimensions of the rectangular garden. Let's assume the length of the garden is x feet.

Now, let's determine the width of the rectangular garden. Since a rectangular garden has two equal lengths and two equal widths, we can express the perimeter of the garden in terms of x:

Perimeter = 2(length + width)
1200 = 2(x + width)

Simplifying the equation:
600 = x + width
width = 600 - x

Next, we can calculate the area of the rectangular garden by multiplying its length and width:
Area = length × width
Area = x × (600 - x)

To find the maximum area, we need to find the value of x that maximizes the Area formula. We can do this by finding the vertex of the quadratic function, considering that the graph of the function is a parabola.

To find the x-coordinate of the vertex, we use the formula:
x = -b / 2a

In our case, the quadratic function is:
Area = -x^2 + 600x

Comparing this function to the general quadratic function form (Ax^2 + Bx + C), we have:
A = -1
B = 600
C = 0

Plugging these values into the formula, we can find the x-coordinate of the vertex:
x = -600 / (2*(-1))
x = -600 / -2
x = 300

Now that we have the value of x, we can find the maximum area of the garden:
Area = x × (600 - x)
Area = 300 × (600 - 300)
Area = 300 × 300
Area = 90,000 square feet

Therefore, the maximum area the garden can have is 90,000 square feet when the length is 300 feet and the width is 300 feet.