Final answer:
To maximize the area with 200 feet of fencing for a rectangular corral, one would create a square with sides measuring 50 feet each.
Step-by-step explanation:
The question asks to find the dimensions of a rectangle corral that would enclose the greatest area, using 200 feet of fence material. This is a maximization problem in the realm of high school algebra and optimization.
To solve this, we can use the perimeter formula for a rectangle (P = 2l + 2w) where l is the length and w is the width, and we know the total perimeter is 200 feet. With this constraint, we can express one dimension in terms of the other (for instance, w = (200 - 2l)/2). Then, we write the area formula (A = l * w) in terms of a single variable and find the maximum value of this function.
When substituted, the area function becomes A(l) = l * ((200 - 2l)/2). To find the maximum area, take the derivative of A with respect to l, set the derivative equal to zero, and solve for l. This will give us the value of l that maximizes the area. Since the problem is symmetrical, the maximum area will occur when l = w, hence for the maximum area, our corral will be a square with sides of length 50 feet.