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Math Help - Optimization/ Max Revenue.

  1. #1
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    Optimization/ Max Revenue.

    Suppose you own an apartment building containing 100 units. If you charge $400 per
    month for each unit, then all units can be rented out. For every $20 increase in monthly rent, you
    will lose one customer. What monthly rent should you charge to maximize your revenue? Show that
    the corresponding revenue is an absolute maximum.


    Attempt: (400+20x)(100-x)
    40,000+1600x-20x^2
    16000-40x
    x=400

    completing the square method:
    (x+40)^2+400

    both yield 400 as answer which would make max rent 8400 which is incorrect. IS my formula wrong? Where did i go wrong
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  2. #2
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    Re: Optimization/ Max Revenue.

    Quote Originally Posted by skg94 View Post
    Suppose you own an apartment building containing 100 units. If you charge $400 per
    month for each unit, then all units can be rented out. For every $20 increase in monthly rent, you
    will lose one customer. What monthly rent should you charge to maximize your revenue? Show that
    the corresponding revenue is an absolute maximum.


    Attempt: (400+20x)(100-x)
    It is hard to say what you might be doing wrong when you don't say what you are doing!
    I see "20x" so is "x" then is the number of "$20 increases"? Then the revenue from each unit is 400+ 20x and the number of apartments rented is 100- x.
    So, yes, the total revenue is (400+ 2x)(100- x)= 40000- 1600x+ x^2[/tex]

    40,000+1600x-20x^2
    16000-40x
    You have an extra "0" here. Although you do not say it, you have taken the derivative of the revenue function. That would be 1600- 40x, not 16000- 40x.

    x=400
    Setting the derivative equal to 0, and solving for x gives x= 40, not 400. Maximum revenue comes when there are 40 "$20 increases in monthly rent" which would mean each apartment rents for 400+ 20(40)= $1200 so that you rent 100- 40= 60 apartments for total revenue 60(1200)= $72000.

    completing the square method:
    (x+40)^2+400
    No, it doesn't. (x+ 40)^2+ 400= x^2+ 80x+ 1600+ 400= x^2+ 80x+ 2000.
    Completing the square on -(20x^2- 1600x+ 4000)= -20(x^2- 80x+ 2000)= -20(x^2- 80x+ 1600)+ 400= (x- 40)^2+ 400.
    The graph of that is a parabola which has vertex at x= 40, y= 400. But it is the "x= 40" that is the number of "$20 increases", NOT the "y= 400".

    both yield 400 as answer which would make max rent 8400 which is incorrect. IS my formula wrong? Where did i go wrong[/QUOTE]
    No, both yield 40, not 400, as the answer which makes the rent on each apartment $1200
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  3. #3
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    Re: Optimization/ Max Revenue.

    omg, so i had it right just a simple mistake.

    And for completing the square thank you for clearing that up
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