Express the area in terms of the length and breadth of the [assumed rectangular] corrals, which should be a quadratic expression.
Quadratic expression has either a maximum or minimum value. In this case we expect a maximum value.
To find this value, one of the way is to by completing the square.
I doubt differentiation is what you have missed out, and I guess you haven't learned yet.
Here, if we assume the 6 corrals are separated, then need to argue that the maximum area would be achieve by constructing the maximum area for each corral using 1/6 of the available fencing.
Or, we can also assume a topology of the 6 corrals to be adjacent to one another in a row, which is more logical/natural and efficient use of the fencing.
Using the latter assumption, let x feet be the length of the row of 6 corrals. The width can be expressed in terms of x by considering 1000-2x = 7 widths, thus the width is (1000-2x)/7.
Let A squared feet be the area of the 6 corrals.
A = x * (1000-2x)/7
= 2(x^2 - 500x)/7
By completing the square:
A = -2/7 * [(x - 250)^2 - 250^2]
Thus, maximum A when (x-250)^2 is minimum, i.e. zero.
length, x = 250 feet, width = (1000-500)/7 feet
and area, A = -2/7 * (- 250^2)
