Let x represent side opposite to highway and y represent other two opposite sides as shown in the diagram.
The perimeter of the parking lot will be sum of 3 sides that is [tex]x+y+y=x+2y[/tex].
We have been given that the plan is to use 440 feet of fencing to fence off the other three sides. This means that perimeter of 3 sides is 440.
[tex]x+2y=440[/tex]
[tex]x=440-2y[/tex]
We know that area of rectangle is length times width, so area of parking lot will be [tex]A=x\cdot y[/tex]
Upon substituting value of x, we will get:
[tex]A(y)=(440-2y)\cdot y[/tex]
[tex]A(y)=440y-2y^2[/tex]
Now we will find the derivative of area function as:
[tex]A'(y)=\frac{d}{dy}(440y)-\frac{d}{dy}(2y^2)[/tex]
[tex]A'(y)=440-4y[/tex]
Let us find critical point by equating derivative to 0.
[tex]440-4y=0[/tex]
[tex]440=4y[/tex]
[tex]\frac{440}{4}=\frac{4y}{4}[/tex]
[tex]110=y[/tex]
Now we will substitute this value is equation [tex]x=440-2y[/tex] to solve for x as:
[tex]x=440-2(110)[/tex]
[tex]x=440-220=220[/tex]
Therefore, the dimensions of 220 feet by 110 feet will enclose the maximum area.
