A college is planning to construct a rectangular parking lot on land bordered on one side by a highway. The plan is to use 560 feet of fencing to fence off the other three sides. What dimensions should the lot have if the enclosed area is to be a​ maximum?

Answer 1

Let's assume the length of the rectangular parking lot is x feet.
Then, the width of the rectangular parking lot will be (560 - 2x) / 2 feet, as there are two equal width sides and the other side is the highway.
The area of the rectangular parking lot is given by length x width, which is x * (560 - 2x) / 2 = (560x - 2x^2) / 2 = 280x - x^2.
To find the maximum area, we take the derivative of the area function and set it equal to zero.
The derivative of the area function is dA/dx = 280 - 2x.
Setting it equal to zero, we get 280 - 2x = 0, or x = 140.
Therefore, the length of the rectangular parking lot is 140 feet.
And the width of the rectangular parking lot is (560 - 2(140)) / 2 = 560 - 280 = 280 feet.
So, the dimensions of the lot that would result in a maximum area are 140 feet by 280 feet.