A rancher has 200 ft of fencing to enclose two adjacent rectangular corrals.
(A) Write the area \(A\) of the corrals as a function of \(x\).
(B) Create a table showing possible values of \(x\) and the corresponding areas of the corral. Use the table to estimate the dimensions that will produce the maximum enclosed area.
(C) Use a graphing utility to graph the area function. Use the graph to approximate the dimensions that will produce the maximum enclosed area.
(D) Write the area function in standard form to find analytically the dimensions that will produce the maximum area.
Answer (2)
(A) To write the area of the corrals as a function of x, we need to determine the dimensions of the two rectangles. Let's assume one rectangle has length x and width y, and the other rectangle has length (200 - x) and width y. The total area A is the sum of the areas of the two rectangles, which is:
A = x * y + (200 - x) * y = xy + 200y - xy = 200y
(B) To create a table of possible values of x and corresponding areas, we can choose different values of x within the given constraints (0 < x < 200) and calculate the corresponding areas using the formula A = 200y. For example:
x = 50, A = 200 * 50 = 10000
x = 100, A = 200 * 100 = 20000
x = 150, A = 200 * 150 = 30000
By observing the table, we can see that as x increases, the area A also increases. Therefore, we can estimate that the dimensions that will produce the maximum enclosed area have x = 200 and y = 0. This means one rectangle will have a length of 200 and the other will have a length of 0.
(C) Using a graphing utility, we can graph the area function A = 200y and observe the trend. The graph will be a line with a positive slope, indicating that as x increases, the area A also increases. The point (200, 0) represents the maximum enclosed area where one rectangle has a length of 200 and the other has a length of 0.
(D) To write the area function in standard form, we can rearrange the equation A = 200y such that y is the subject.
Dividing both sides by 200, we get:
y = A/200
This form allows us to find the dimensions analytically by substituting the value of A into the equation. For maximum area, A will be equal to the maximum value obtained from the table, which is 30000. Substituting A = 30000 into the equation, we get:
y = 30000/200 = 150
Therefore, the dimensions that will produce the maximum enclosed area are x = 200 and y = 150.
The area of two adjacent rectangular corrals, given a fencing of 200 ft, can be expressed as a function of width x as A(x) = 200x - 2x². A table and a graph show that the maximum area occurs at x = 50 ft, with the dimensions leading to the area of 5000 sq ft. Rewriting this function confirms that the maximum area is achieved with corrals of 50 ft wide and 100 ft long.
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