text: # Enclosing a Rectangualr Field
  
  • Oct 7th 2005, 11:00 AM
  Robthebear
  Enclosing a Rectangualr Field
  Beth has 3000 feet of fencing available to enclose a rectangular field.
  
  a) Express the area A of the rectangle as a function of x, where x is the length of the rectangle.
  
  b) For what value of x is the area largest?
  
  c) What is the maximum area?
  
  I have the answer but do not know how to set up. Can someone help.
  
  a) A(x)=-x^2 + 1500x
  
  b) A is largest when X = 750ft.
  
  c) 565,500 ft sq
  
  Robert
  • Oct 7th 2005, 11:34 AM
  ticbol
  One way of doing this is by Analytic Geometry, by the properties of the parabola.
  Another way, an easier way, is by Calculus. I will use this way here.
  
  Rectangle.
  Length = x ft.
  Perimeter = 3000 ft.
  So,
  Width = (3000 -2x)/2 = 1500 -x ft.
  
  a) Area, A = x(1500 -x) = 1500x -x^2
  
  b) A is largest or smallest when dA/dx = 0
  dA/dx = 1500 -2x
  Set that to zero,
  0 = 1500 -2x
  2x = 1500
  x = 1500/2 = 750 ft.
  That should be the x for maximum A. (because minimum A is zero, and here x=0 or x=1500 ft.)
  
  c) A = 1500x -x^2
  max A = 1500(750) -(750)^2 = 562,500 sq.ft.
  (Not 565,500 sq.ft.)