Final answer:
To find the maximum area of a rectangular corral with three equal-sized sections and a perimeter of 230 feet, the area in terms of x is A = x * (230 - 4x) / 2, with x's domain being (0, 57.5). Using a calculator to maximize this function, the optimal width x is approximately 28.75 feet.
Step-by-step explanation:
The problem involves creating a rectangular corral with three equal-sized sections using 230 feet of fencing material. To express the total area A in terms of the width x, first remember that the perimeter P is the sum of all sides. If the corral is divided into 3 sections, there would be 4 lengths of x and 2 widths making up the total perimeter, so 4x + 2y = 230, where y is the length of the corral. Since the sections are equal in size, the total area can be given as A = x * y. But we need to express y in terms of x from our perimeter equation which gives us y = (230 - 4x) / 2. Thus, the area equation in terms of x is A = x * (230 - 4x) / 2.
The domain of x is the set of all possible values that x can take. Since x has to be a positive number (you can't have negative distance) and we have 230 feet of fencing creating 4 lengths of x, the maximum value of x would be 230/4, or 57.5 feet. Hence, the domain of x is (0, 57.5).
To find the dimensions which produce the maximum area, use calculus or a calculator to find the maximum point on the graph of the area function. After using a calculator with optimization functionality (or calculus techniques such as finding the derivative and setting it to zero), we'll find that x is approximately 28.75 feet (rounded to three decimal places), and y would then be equal to (230 - 4*28.75)/2. You can verify by calculation that these dimensions would give you the maximum area.