A college is planning to construct a rectangular parking lot on land bordered on one side by a highway. The plan is to use 680 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?

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wagonbelleville wagonbelleville · Answer 1 · 2019-02-14T06:25:30+01:00

Answer:

The maximum length is 170 feet and maximum width is 340 feet.

Step-by-step explanation:

Let the length of the parking = x feet and width = y feet.

Since, the fencing is to be applied on three sides of the lot.

So,we get, [tex]2x+y=680[/tex]

i.e. [tex]y=680-2x[/tex]

Since, the area of the rectangular lot = Length × Width

So, area of the lot = [tex]x \times y[/tex]

i.e. area of the lot = [tex]x \times (680-2x)[/tex]

i.e. area of the lot = [tex]680x-2x^2[/tex]

Now, we know the maximum value of 'x' in [tex]ax^2+bx+c[/tex] is [tex]\frac{-b}{2a}[/tex].

So, the maximum length of the lot is x= [tex]\frac{-680}{2\times -2}[/tex] = [tex]\frac{680}{4}[/tex] = 170 feet.

Then, [tex]y=680-2x[/tex] i,plies [tex]y=680-2\times 170[/tex] implies [tex]y=680-340[/tex] i.e. y= 340 feet.

So, the maximum length is 170 feet and maximum width is 340 feet.