find maximum area of the parking lot if you only have 2,9000 feet

Question

Jim wants to build a rectangular parking lot along a busy street but only has 2,900 feet of fencing available. If no fencing is required along the street, find the maximum area of the parking lot

Answer 1

Using 2900 sq. ft of fencing, you have 2 sides that we will label x and the 3rd side is 2900 - 2x.  Now we have three sides.  Using area formula we have 2900 = x(2900-2x)= 2900x-2x^2.  Using commutative

property we have -2x^2 +2900x  which is a quadratic. a=-2, b=2900 and c=0.

Maximum area is where x= -b/2a or -2900/2(-2)=725 area = (2900*725)-2(725^2)=1,051,250 sq ft.

Answer 2

I don't know what level of math you are in so we'll solve the problem two ways.

P=2l+2w

2900=2l+2w

1450=l+w

1450-w=l

A=lw

A=(1450-w)w

A=1450w-w²

use -b/2a

-1450/-2=725

you want a square with each side measuring 725 feet.

However, no fencing is required along the street.

Make each width be 725 feet and make the length be 725 feet plus the other 725 feet that would have been used on the street side for a length of 1450 feet.

The dimensions are 725x1450 for the rectangular parking lot.

A=lw

A=725*1450=1,051,250 square feet. This is the maximum area.

Notice that the rectangle is made up of two squares 725x725 each.

 

Using simple arithmetic...

The maximum area is always a square.

2900/4=725

One side of the square is not needed.

Add the 725 feet not needed to the opposite side to get 1450 for the length and the two widths would be 725 each.

The rectangle again is 725 by 1450 and the area is 725*1450=1,051,250 square feet.

 

Answer 3

Alexa,

 

Sorry, I run through this so fast and I made a mistake

x = 29,000/4 = 7,250 feet is the side that extends from the street

and 29,000 - 2(7,250) = 14,500 feet is the side that runs parallel with the street

 

You will, hopefully, do enough of this to know the answer right away - for a fixed perimeter, a square gives you the maximum area. Here it is

 

Your three sides are x, x, and 29,000 - 2x

 

Area = x(29,000 - 2x) = 29,000x - 2x²

 

Take the derivative and you get 29,000 - 4x

 

You set this to zero because it is a downward-opening that you have for area and it's slope is zero at that maximum

 

29,000 - 4x = 0

x = 29,000/4

I hope this helps