A farmer wants to make 9 identical rectangular enclosures as shown in the diagram below. If he has 720 feet of fencing materials, what should the dimensions of each enclosure be if the total area is to be maximized?

To maximize the total area of the 9 identical rectangular enclosures, we need to find the dimensions of each enclosure that will use up the entire 720 feet of fencing materials.

Let's assume the width of each enclosure is "w" and the length is "l".

Since there are 9 identical enclosures, each enclosure will have a length of l and a width of w.

The total fencing required for each enclosure can be calculated as follows:
Perimeter of each enclosure = 2w + 2l

Since we have 9 enclosures, the total fencing required for all enclosures will be:
Total fencing = 9 * (2w + 2l)

Now we can equate the total fencing with the total amount available:
9 * (2w + 2l) = 720

Dividing by 9, we get:
2w + 2l = 80

Simplifying this equation, we have:
w + l = 40

To maximize the total area, we need to maximize the length and width.

To do this, we can set w = l, as this will result in a square shape, which maximizes the area for a given perimeter.

Substituting w = l in the equation w + l = 40, we have:
2l = 40

Dividing by 2, we get:
l = 20

So the length of each enclosure should be 20 feet.

Since we assumed w = l, the width of each enclosure will also be 20 feet.

Therefore, the dimensions of each enclosure should be 20 feet by 20 feet to maximize the total area.

To find the dimensions of each enclosure that will maximize the total area, we need to set up an optimization problem.

Let's denote the length of each enclosure as L and the width as W. We know that each enclosure is a rectangle, so the perimeter of each enclosure can be represented as:

2L + 2W = perimeter

Given that the farmer has a total of 720 feet of fencing materials, we have:

2L + 2W = 720

We also know that the total area of all 9 enclosures should be maximized. The area of each enclosure is given by:

A = L * W

Since there are 9 enclosures, the total area can be represented as:

9A = 9LW

To maximize the total area, we need to find the dimensions that satisfy the perimeter constraint while also maximizing the value of A.

Next, we can solve this problem using calculus. We need to express one variable in terms of the other from the perimeter equation. Let's isolate L:

2L + 2W = 720
2L = 720 - 2W
L = 360 - W

Now substitute L in terms of W into the area equation:

9A = 9LW
9A = 9(360 - W)W
9A = 3240W - 9W^2

To find the maximum area, we take the derivative of A with respect to W and set it equal to zero:

dA/dW = 3240 - 18W = 0
18W = 3240
W = 3240/18
W = 180

Now substitute this value of W back into the perimeter equation to find L:

2L + 2W = 720
2L + 2(180) = 720
2L + 360 = 720
2L = 720 - 360
2L = 360
L = 180

Therefore, each enclosure should have dimensions of 180 feet by 180 feet to maximize the total area.