completing the square parabola

Question

a science museum is putting up a restaurant along one wall of the museum. Restaurant space will be rectangular. Assume the museum wants to maximize the area for the restaurant. there is 120 feet of fencing available for three sides that require fencing How long will the longest side of the restaurant be? What is maximum area?

Answer 1

P=2l+2w

A=lw

120=2l+2w

60=l+w

60-l=w

A=lw

A=l(60-l)

A=60l-l²

-b/2a=-60/2(-1)=-60/-2=30

you would have a square 30 feet by 30 feet (a square always gives the maximum area so you could have just divided 120 by 4 to get 30 feet on a side)

however, one side does not need fencing so you have 30 feet to add to the opposite side of the wall of the museum

you have a rectangle that is 30 feet by 60 feet (30+30) by 30 feet by "the wall"

the maximum area is 30 feet*60 feet=1800 square feet

notice that this rectangle is actually 2 30x30 squares put together (maximum area is always a square !)