Final answer:
The total area A of two adjacent rectangular corrals can be represented as a function of x using the formula A(x) = x*(400 - 3x) / 2. To find the dimensions for maximum area, we differentiate A(x) with respect to x, set it to zero, and solve for x, which is approximately 66.67 feet for both length and width, indicating that two squares are formed.
Step-by-step explanation:
A rancher has 400 feet of fencing to enclose two adjacent rectangular corrals. To write the total area A of the corrals as a function of x, let's assume that the length of the corrals is x feet and the width is y feet. Since there are three lengths of x, and two widths of y, which should add up to 400, we can formulate the equation as 3x + 2y = 400. We can express y in terms of x using y = (400 - 3x) / 2. The total area A, which is equal to the sum of the areas of the two corrals, is x times y, so A = x*y, which simplifies to A = x*(400 - 3x) / 2.
To find the dimensions that produce a maximum enclosed area, we differentiate the area function A(x) with respect to x, set the derivative to zero, and solve for x. This will give us the value of x that maximizes the area. Applying this to the area function A(x) = x*(400 - 3x) / 2, we find the derivative, set it to zero and solve for x. The calculation shows the maximum area is attained when x is equal to 400/6 or approximately 66.67 feet. The corresponding width y would then be (400 - 3*66.67) / 2, which is approximately 66.67 feet as well, producing two squares with maximum area.