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Question

A rancher is going to build three sides of a rectangular corral next to a river. He has 240 feet of fencing and wants to enclose the maximum area possible inside the corral. What are the dimensions of the corral with the maximum area and what is the area? 

Answer 1

P=2l+2w

P=l+2w because of the river

240=l+2w

240-2w=l

A=lw

A=(240-2w)(w)

0=240w-2w^2

this is a parabola opening downward so it has a maximum point, (h,k), where h=-b/2a which gives you the "maximizing" point

-b/2a=-240/-4=60 feet

ordinarily you would have a 60 by 60 by 60 by 60 square but one side is not needed

add the 60 feet not needed to the opposite side to get 120 feet

you have a 60 by 120 by 60 by "the river" rectangle

A=60*120=7200 square feet  (answer)

notice that this rectangle is really two 60 by 60 squares adjacent to each other