Expert: Socrates - 11/26/2010

Question
the sudbary community centre wants to enclose a rectangular swimming area at a public beach on the shores of ramsey lake. the city has provided 300 m of rope and buoys to mark off the area. what is the maximum swimming area (in m^2) that can be enclosed along the beach with the 300 m of rope? i understand the question, but i can't figure out how to get the answer :S PLEASE HELP ME!

Answer
You need to draw a picture of this

Let W be the width of the swimming area
Let L be the length

Since one end of the rectangular swimming area will be the beach , there are only 3 sides to be bordered by the rope.

The total amount of rope used will be 300 = 2L+W

The area is LW

300 = 2L+W so W = 300-2L


area is then (L)(300-2L) = 300L - 2L^2



So we want the maximum value for the quadratic 300L - 2L^2

The maximum occurs at the average of the roots

The roots , or zeros for 300L - 2L^2 = (L)(300-2L) are L = 0 and L=150

The average is (0+150)/2 = 75

Substitute 75 for L in the expression for area

Maximum area = (75)(300-(2)(75))= (75)(150) = 11,250 square meters