The back of Dante's property is a creek. Dante would like to enclose a rectangular area, using the creek as one side and fencing for the other three sides, to create a corral. If there is 200 feet of fencing available, what is the maximum possible area of the corral?
length = x
width = w
x + 2 w = 200 so x = 2 (100-w)
A = x w = 2 (100-w)w
A = 200 w - 2 w^2
now I do not know if you do calculus but if you do
dA/dw = 0 at max = 200 - 4w
w = 50
x= 2(50) = 100
A = 5,000
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if you do not do calculus find vertex of parabola
A = 200 w - 2 w^2
w^2 -100 w = - A/2
w^2 -100 w + 50^2 = -A/2 +2500
(w-50)^2 = -1/2(A-5000)
vertex at w = 50 and A = 5,000 as we already knew from the calculus
To find the maximum possible area of the corral, we need to determine the dimensions of the rectangle that will maximize the area while utilizing the given amount of fencing.
Let's assume the length of the corral is parallel to the creek (which is the side formed by the creek) and the width is perpendicular to the creek.
Let's set up the formula to calculate the area of the corral:
Area = length * width
Given that the creek forms one side of the corral, we already have the length of the corral as the length of the creek. However, we need to figure out the values for the width and the two remaining sides.
Since there are three fence sides, and the length of the creek side is already accounted for, we have two equal-length fence sides left to consider.
Let's denote the length of each of these two fence sides as "x". Therefore, the total length of these two fence sides is 2x.
We also know that the total length of the fencing available is 200 feet.
So we have the equation: 2x + length of the creek side = total length of fencing
2x + length of creek = 200
Since the length of the creek side is not given, we cannot solve this equation directly. However, we can still find the maximum possible area by considering the fact that x should be as large as possible.
To maximize x, we can assume that the fences are placed as closely as possible to the creek. In this case, the width of the corral (perpendicular to the creek) will be 0.
So, the maximum possible width is 0, and the corresponding length of the corral (parallel to the creek) will be the entire length of the fencing, which is 200.
Now, we can calculate the maximum possible area of the corral:
Area = length * width
Area = 200 * 0
Area = 0 square feet
Therefore, the maximum possible area of the corral is 0 square feet when the fences are placed as closely as possible to the creek.