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.)
explanation: The document contains a math problem related to maximizing the area of a rectangular field given a fixed amount of fencing. The problem is broken down into three parts, and the user provides both the question and an attempted answer. The response from another user corrects and explains the solution.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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.)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem related to maximizing the area of a rectangular field given a fixed amount of fencing. The problem is broken down into three parts, and the user provides both the question and an attempted answer. The response from another user corrects and explains the solution.
  
  Label: question-and-answer