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Maximizing Profit and such

  1. Feb 24, 2010 #1
    1. A company makes 2 products. Desk and Bookcase. The company has 2 factories and each product requires development time in both factories. Factory A only has 55 hours a week, and factory B only has 39 hours. To make a desk it takes 4 hours in factory A and 3 hours in factory B. To make a bookcase it takes 3 hours in factory A and 2 hours in factory B. Profits for the desk are 70 dollars, and 50 dollars for the bookcase.

    That is all the information available...There are multiple questions, but one of the questions I encountered that I for whatever reason cannot coherently attempt is "Write down the profit function for the sale of X desks and Y bookcases."

    2. It also asked to make 2 ineqaulities for the hours in each factory..which I think I did correctly?


    3. P=x(70)+y(50)
  2. jcsd
  3. Feb 25, 2010 #2


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    If you make X desks and make a profit of 70 dollars on each, you make 70X dollars profit on those desks. If you make Y bookcases and make a profit of 50 dollars on each, you make 50Y dollars profit on those bookcases. Together, you make 70X+ 50Y dollars profit.
    That is NOT the hardest part of these kinds of problems!

    First, there are no "a" or "b" or "x" or "y" in your problem. They did label the factories "A" and "B" and they say "X desks" and "Y book cases" but "a", "b", "x", and "y" are NOT the same as "A", "B", "X", and "Y".

    Also, if you mean "4A+ 3B<= 55", that makes no sense. A is a factory, not a number. You cannot multiply a number by a factory!

    In other words, write down what letters you are using to represent what numbers and be consistent.

    If X is the number of desks and Y is the number of book cases made, then since "To make a desk it takes 4 hours in factory A and 3 hours in factory B" and "To make a bookcase it takes 3 hours in factory A and 2 hours in factory B", with X desks and Y book cases, it will take 4X+ 3Y hours in factory A. Since factory A is restricted to 55 hours:
    4X+ 3Y<= 55./

    It will take 3X+ 2Y hours in factory B and factory B is restricted to 39 hours, so
    3X+ 2Y<= 39.

    The problem should eventually be "how many desks and bookcases should you make to maximize your profit?" The key idea in "linear programming" is that a linear function, like the profit 70X+ 50Y, on a convex polygon will take on both maximum and minimum values at the vertices of the polygon.

    Here, since the "feasible region" is given by the inequalities X>= 0, Y>= 0, 4X+ 3Y<= 55, and 3X+ 2Y<= 39, the boundaries of that region are the lines X= 0, Y= 0, 4X+ 3Y= 55, and 3X+ 2Y= 39. Find the points where those lines cross, and evaluate 70X+ 50Y at each point to see which is largest.
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