To find the dimensions of the rectangular parking lot that will maximize the enclosed area, we can use the concept of optimization.
Let's assume the length of the parking lot is L (in feet) and the width is W (in feet). Since the parking lot is bordered on one side by a highway, the highway side doesn't require any fencing. The other three sides will be fenced, which will use up the 680 feet of fencing available.
Given that the perimeter of a rectangle is calculated by adding up all its side lengths, we can determine the perimeter of the parking lot as follows:
Perimeter = 2L + W
Since the available fencing is 680 feet, we can write the equation:
2L + W = 680 .....(Equation 1)
Now, let's consider the area of the parking lot. The area of a rectangle is calculated by multiplying its length by its width:
Area = L * W
In order to maximize the enclosed area, we need to find the maximum value of Area. To do this, we can solve for one variable in Equation 1 and substitute it into the area equation.
Rearrange Equation 1 to solve for W:
W = 680 - 2L
Substitute this value of W into the area equation:
Area = L * (680 - 2L)
Area = 680L - 2L^2
Now we have the area in terms of L alone. To find the maximum area, we can take the derivative of the area equation with respect to L, set it equal to zero, and solve for L.
d(Area)/dL = 680 - 4L
Setting the derivative equal to zero:
680 - 4L = 0
4L = 680
L = 170
So, L = 170. We can substitute this value back into Equation 1 to find the width:
2L + W = 680
2 * 170 + W = 680
340 + W = 680
W = 340
Therefore, the dimensions that maximize the enclosed area would be a length of 170 feet and a width of 340 feet.