The back of Alisha’s property is a creek. Alisha would like to enclosure a rectangular area, using the creek as one side and fencing for the other threee sides, to create a corral. If there is 180 feet of fencing available, what is the maximum possible area of the corral?
Area=xy
perimenter= 2x+y
or y=180-2x
area=x(180-2x)
now area is zero at x=0, and x=90
so max area must be half way between those roots, or x=45
Think about that, as area is a parabola. Max area is 1/2 way between roots.
Areamax=xy=45(180-90)=....
To find the maximum possible area of the corral, we need to determine the dimensions of the rectangle that will maximize the area given the constraints.
Let's assume the length of the side parallel to the creek is "x" feet, and the other two sides (perpendicular to the creek) have lengths "y" feet each.
Since the creek acts as one side of the corral, we only need to fence the other three sides. So, the total length of the fencing used is given by:
Length of fencing = 2y + x
According to the given information, we have 180 feet of fencing available. So, we can write the equation:
2y + x = 180 ----(1)
Now, we need to express the area of the corral in terms of "x" and "y". The area of a rectangle is given by:
Area = length * width
In this case, the length is "x" and the width is "y". Hence, the area is:
Area = x * y ----(2)
To find the maximum possible area, we need to solve equations (1) and (2) simultaneously.
From equation (1), we can express "x" in terms of "y" as:
x = 180 - 2y
Substitute the expression for "x" in equation (2):
Area = (180 - 2y) * y
Simplifying this equation:
Area = 180y - 2y^2
Now, we have expressed the area as a function of "y". To maximize the area, we can differentiate the equation with respect to "y" and set it equal to zero:
d(Area)/dy = 180 - 4y = 0
Solving for "y":
180 - 4y = 0
4y = 180
y = 45
Now that we have the value of "y", we can substitute it back into equation (1) to find the corresponding value of "x":
2(45) + x = 180
90 + x = 180
x = 90
Therefore, the maximum area of the corral is achieved when the dimensions are:
Length (x) = 90 feet
Width (y) = 45 feet
Substituting these values into equation (2) gives us the maximum area:
Maximum Area = (90)(45) = 4050 square feet.