A college is planning to construct a rectangular parking lot on land bordered on one side by a highway. The plan is to use 400 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?

Question

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Part 1
The width is enter your response here feet. The length is enter your response here feet. ​(Let the longer side be the length and the shorter side the​ width.)

Answer 1

To find the dimensions that will maximize the enclosed area, we can use the formula for the area of a rectangle: A = length * width.

Let's assume the width is x feet. The length, which is the side bordering the highway, will then be 400 - 2x feet (since there are three sides to fence off and 400 feet of fencing total).

Now we can substitute these values into the formula for the area:

A = (400 - 2x) * x
A = 400x - 2x^2

To find the maximum area, we can take the derivative of this function and set it equal to zero:

dA/dx = 400 - 4x
400 - 4x = 0
4x = 400
x = 100

Therefore, the width of the parking lot should be 100 feet. The length, which is the side bordering the highway, will be 400 - 2(100) = 400 - 200 = 200 feet.

So, the dimensions for the lot that will maximize the enclosed area are: width = 100 feet and length = 200 feet.