Final answer:
To maximize the area of a rectangular field with 1248ft of fence, we can derive a formula in terms of the field's length and then use calculus to find the field length that maximizes the area. The maximum area is found by substituting this length into the area formula.
Step-by-step explanation:
The problem involves determining the maximum area of a rectangular field that can be fenced using a given length of fencing. To find the maximum area of the field that a farmer can obtain with 1248ft of fence, while dividing the field in half with a fence down the middle parallel to one side, we can use a little bit of algebra and calculus.
Let's denote the length of the rectangular field as L and the width as W. The total length of the fence includes the outer perimeter and the fence dividing the field, which is 2W + 3L (since there would be one length (L) in the middle and two widths (W) on the sides). So, the equation for the perimeter is:
2W + 3L = 1248
Since the farmer wants the maximum area, we are trying to maximize the function A = L * W. We can express W in terms of L from the perimeter's equation:
W = (1248 - 3L) / 2
Substitute W into the area equation:
A = L * ((1248 - 3L) / 2)
To find the maximum value of A, we would take the derivative of A with respect to L, set it to zero, and solve for L. After finding the value of L that maximizes the area, we would then find that value of W. The product of L and W would give us the maximum area of the rectangular field that the farmer can fence.