The owner of a farm wants to form four rectangular corrals. He has 750 meters of iron gate to enclose and to separate the four corrals. To save materials, he he decides to enclose a large area and to divide into four rectangles. Divide them will use part of the 750 meters of iron gate. Which are the dimensions of the largest rectangular area than can enclose and to separate to form the four corrals?

To find the dimensions of the largest rectangular area that can enclose and separate into four corrals, we can use the concept of maximizing the area while minimizing the perimeter.

Let's assume the dimensions of the rectangular area are length (L) and width (W) with L > W. The perimeter of the rectangular area would be P = 2L + 2W. We want to maximize the area (A) while using only 750 meters of iron gate.

The owner wants to divide the enclosure into four corrals, which means there will be three partitions, each requiring one length of iron gate. So, the total perimeter used for the enclosure and partitions would be P_total = P + 3L.

Given that P_total = 750 meters, we have:

750 = P + 3L
750 = 2L + 2W + 3L

Simplifying the second equation:
750 = 5L + 2W

We need to express W in terms of L to have only one variable in the equation. Let's solve the second equation for W:

W = (750 - 5L) / 2

Now, substitute this expression for W in the area equation:

A = LW = L * (750 - 5L) / 2

To maximize the area A, we can take the derivative of A with respect to L, set it to zero, and solve for L:

dA/dL = (750 - 10L) / 2 = 0
750 - 10L = 0
10L = 750
L = 75

Now, substitute the value of L back into the equation for W:

W = (750 - 5(75)) / 2
W = (750 - 375) / 2
W = 375 / 2
W = 187.5

Therefore, the dimensions of the largest rectangular area that can enclose and separate into four corrals are:

Length (L) = 75 meters
Width (W) = 187.5 meters

Note: Since the dimensions are irregular (75 and 187.5), you may want to round them to practical values.