Ethan wants to build a rectangular enclosure for his animals.

Question

Ethan wants to build a rectangular enclosure for his animals. One side of the pen will be against the barn, so he needs no fence on that side. The other three sides will be enclosed with wire fencing. If Ethan has 550 feet of fencing, you can find the dimensions that maximize the area of the enclosure.

a) Let w be the width of the enclosure (perpendicular to the barn) and let l be the length of the enclosure (parallel to the barn). Write an function for the area A of the enclosure in terms of w. (HINT first write two equations with w and l and A. Solve for l in one equation and substitute for l in the other).

b) What width w would maximize the area?

c) What is the maximum area?

Answer 1

 

Length is L

width is w

 

Perimeter = L + 2w <---- only one length

550 = L + 2w

550 - 2w = L

 

Area = length * width

 

         = (550 - 2w) w

 

 

THe max occurs at the AVERAGE of the solutions to this

quadratic parabola.

 

The solutions are:

550 - 2w = 0

550 = 2w

w = 550/2 = 275

 

and

 

w=0

 

The average of these solutions are

 

(275+0)/2 = 137.5

 

The dimensions needed for max area are

width w=137.5 and length L=550 - 2(137.5)

                                         = 550 - 275

                                          = 275

 

The max area is 275 * 137.5 = 37812.5