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Math Help - Objective functions, region of feasible solutions

  1. #1
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    Objective functions, region of feasible solutions

    Hi this is my first time here and was searching on the web (google) for some math help. My math teacher is giving us a test later this week and gave us some practice problems. I was wondering if anyone could help me with one of these problems.

    Here is the question:

    Sam and Doris manufacture rocking chairs and porch swings in the western Upper Peninsula. The times (in hours) each of them needs to spend assembling each rocker and swing are shown below. Also shown are the total amounts of time per week each has available, together with the sale price of each chair and each rocker.


    Sam: chair: 3 Porch Swing: 2 Maximum Available: 48


    Doris: chair: 2 Porch Swing: 2 Maximum Available: 40


    Unit Price: Chair: $160 Porch Swing: $100



    Let
    x denote the number of rocking chairs to produce per week in order to maximize revenue


    y denote the number of porch swings to produce per week in order to maximize revenue

    (a) Write out the objective function and the constraints for this problem.

    (b) Graph the region of feasible solutions and completely solve the problem.


    Last edited by Cazper; April 22nd 2007 at 03:08 PM.
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  2. #2
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    Hello, Cazper!

    Sam and Doris manufacture rocking chairs and porch swings.
    The times (in hours) each of them needs to spend assembling
    each chair and swing are shown below.
    Also shown are the total amounts of time per week each has available,
    together with the sale price of each chair and each swing.
    Code:
          | Chair | Swing | Max
    - - - + - - - + - - - + - - +
     Sam  |   3   |   2   |  48 |
    - - - + - - - + - - - + - - +
     Dora |   2   |   2   |  40 |
    - - - + - - - + - - - + - - +
    Price |  160  |  100  |
    - - - + - - - + - - - +
    Let x = number of rocking chairs to produce per week.
    Let y = number of porch swings to produce per week.

    (a) Write out the objective function and the constraints for this problem.
    Objective function (revenue): .R .= .160x + 100y

    Constraints
    . . Sam: -3x + 2y .< .48
    . . Dora: .2x + 2y .< .40


    (b) Graph the region of feasible solutions and completely solve the problem.
    First, we have: .x > 0, y > 0. .The region is in quadrant 1.

    Graph the line: .3x + 2y .= .48
    . . It has intercepts: (16,0), (0,24).
    Sketch the line and shade the region below it.

    Graph the line: .2x + 2y .= .40
    . . It has intercepts: (20,0), (0,20).
    Sketch the line and shade region below it.
    Code:
            |
          24*
            | *
            |   *
      (0,20)o     *
            |:::*   *
            |:::::::* * (8,12)
            |:::::::::::o
            |:::::::::::::* *
            |:::::::::::::::*   *
            |:::::::::::::::::*     *
            |:::::::::::::::::::*       *
        - - o - - - - - - - - - - o - - - - * - -
          (0,0)                (16,0)       20
    We have a four-sided figure.
    We are concerned with the coordinates of its vertices.
    Three of the vertices are: (0,0), (16,0) and (0,20)

    The fourth vertex is the intersection of the two slanted lines.
    The intersection of 3x + 2y = 48 and 2x + 2y = 40 is: (8,12)

    Test the vertices: .(x,y) .= .(0,0), (16,0), (20,0), (8,12) in the revenue function
    . . and see which one produces maximum revenue.

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