4.6: Applied Optimization

Applied Optimization is pretty tough,
finishing the homework has been rough.
But this topic is prevalent in many fields,
so I should probably keep my dislike concealed.

Problem:

#9. Suppose 600 ft. of fencing are used to enclose a corral in the shape of a rectangle with a semi-circle whose diameter is a side of a rectangle. Find the dimension of the corral with the maximum area.

Perimenter=2x+y+(pi*y)/2
600=2x+y+(pi*y)/2
x=(600-y-(pi/2)*y)/2

A(x)=(xy+pi(y/2)^2)/2
      =(600y-(y^2)+pi/2*y^2)/2 + 2+(pi*y)/8
      =300y-(y^2)/2- (1/8)*pi*y^2

A'(x)= 300 - (2y/2)-(1/4)*pi*y
A'(x)= 300-y-(1/4)*pi*y =0

y=1200/(4+pi)
x=(600-y-(pi/2)*y)/2

  =600/(4+pi)

So in order to get the maximum amound of space,  the dimensions of the corral need to be (600/(4+pi), 1200/(4+pi))