We are going to fence in a rectangular field and have a maximum of 200 feet of material to construct the fence. Determine the dimensions of the field that will enclose the maximum area?

a square encloses the maximum area for a given perimeter.

To determine the dimensions of the field that will enclose the maximum area, we can use the concept of derivatives.

Let's assume the length of the rectangular field is x feet, and the width is y feet.

Since we have a maximum of 200 feet of material to construct the fence, the total amount of material used for the fencing will be the sum of all the sides of the rectangle:

2x + 2y = 200

Now, we want to find the dimensions that will enclose the maximum area. The area of a rectangle is given by A = x * y.

To find the maximum area, we need to maximize A with respect to x and y. We can rewrite the constraint equation as:

2y = 200 - 2x
y = 100 - x

Substituting this value of y in terms of x into the area equation, we get:

A = x * (100 - x)
A = 100x - x^2

To find the maximum value of A, we need to find the critical points of the function A = 100x - x^2. Taking the derivative of A with respect to x and setting it to zero:

dA/dx = 100 - 2x = 0

Solving this equation, we find x = 50.

Substituting x = 50 back into the constraint equation, we find y = 100 - 50 = 50.

Therefore, the dimensions of the field that will enclose the maximum area are 50 feet by 50 feet.