A rancher with 7000 yds of fencing wants to enclose a rectangular field that borders a straight highway and then wants to devide it into two plots with a fence parellel to the highway. If no fence is needed along the highway, what is the largest area that the farmer can enclose?
the highway doesnt need fence its already fenced can someone pls help?
To find the largest possible area that the rancher can enclose, we need to optimize the dimensions of the rectangular field.
Let's consider the rectangular field with length L and width W. We are given that the rancher has 7000 yards of fencing available.
The perimeter of the rectangular field is given by the equation:
Perimeter = 2L + W
Since the rancher only needs to fence three sides (the two lengths of the rectangle and one width), we have:
Perimeter = 2L + W = 7000
We want to maximize the area A of the rectangular field, given by the equation:
Area = L * W
To solve this problem, we need to express one variable (either L or W) in terms of the other variable. Let's solve the perimeter equation for W:
2L + W = 7000
W = 7000 - 2L
Now we can substitute this expression for W in terms of L into the area equation:
Area = L * (7000 - 2L)
Area = 7000L - 2L^2
To find the maximum area, we need to find the value of L that maximizes the area. One way to do this is by considering the vertex of the parabola given by the area equation.
The vertex of a parabola in the form f(x) = ax^2 + bx + c is given by the x-coordinate x = -b/2a. In our case, the parabola equation is Area = -2L^2 + 7000L.
The coefficient of L^2 is -2, and the coefficient of L is 7000. So, the x-coordinate of the vertex is:
L = -7000 / (2*(-2))
L = -7000 / (-4)
L = 1750
Now, substitute this value of L back into the equation for W:
W = 7000 - 2(1750)
W = 7000 - 3500
W = 3500
So, the dimensions of the rectangular field that maximize the area are L = 1750 yards and W = 3500 yards.
Finally, we can calculate the maximum area by substituting the values of L and W into the area equation:
Area = L * W
Area = 1750 * 3500
Area = 6,125,000 square yards
Therefore, the largest area that the rancher can enclose is 6,125,000 square yards.