pre calc question

Question

 2. A farmer has 150 feet of fencing he is using to create a lot for his pack of llamas. he wants to keep them secluded in this fence which boarders his barn so they can no longer spit on his wife’s flowers. what is the maximum area of the lot he can make for them? what are the dimensions of the lot?

Answer 1

P=l+2w

150=l+2w

150-2w=l

A=lw

A=(150-2w)w

A=150w-2w²

this is a parabola that opens downward so there is a maximum point, (h,k), the vertex

h is the "maximizing" point and k is the maximum area

h=-b/2a

h=-150/(-4)

h=37.5 feet

each width is 37.5 feet and the length is 37.5+37.5=75 feet and the other length is the barn

the maximum area is A=37.5*75=2812.5 square feet

note: the maximum area is always a square

150/4=37.5

because you don't need one side add the 37.5 that you don't need to the opposite side to get 75 

you have a rectangle 37.5 feet by 75 feet by 37.5 feet by "the barn"