A word problem about paraboloas and their applications

Question

A college is planning to construct a rectangular parking lot on land bordered on one side by a highway. The plan is to use 640 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 L = length

Let W = width

*  means multiply

 

Area = L*W

Perimeter= 640 = 2W + L (For this problem since we don't need fence for the side bounded by the highway.)

 

Use systems of equations, solve the perimeter for L and then plug it back in:

 

First I solve for L: 

640 = 2W- L

L=640-2W

 

Then I replace the L in the area equation with  640-2W:

Area = W * (640-2W) = -2W ^2 + 640W

 

Ok now think about this, we want the greatest width and length. The parabola representing this equation is going to be facing downward. (We know this because of the negative 2)

 

Keep in mind the parabola shows the relationship between area and width. So the top of it (the vertex) will give us the highest width and the corresponding highest area. So what is the vertex of the parabola?

 

Width= x = -b/2a = -640/2(-2) = 160 (This tells us the width is 160.)

 

Area = f(x) = -2(160) ^ 2 + 640(160) = -51200 + 102400 = 51200 (This tells us the area would be 51200)

 

In order to find the length we will take the width of 160 and plug it back into one of our earlier equations.

 

640 = 2w + L

 

640 = 2(160) + L

 

L = 320

 

So our best width and length to use would be 160 and 320.  *bows* We have conquered.

 

Alright, time for breakfast, have a great Saturday!