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Math Help - Equation to Represent Cost

  1. #1
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    Question Equation to Represent Cost

    Hello!

    I have to answer a question that reads as follows:

    A real-estate firm owns 100 garden-type apartments. At $400 per month, each apartment can be rented. However, for each $10 per month increase, there will be two vacancies with no possibility of filling them. Let the monthly rent per apartment be x, with x >= 400

    (a) How many apartments are rented if the rent x = 450 ?
    (b) Find a general expression of the number of apartments rented n=n(x) if the rent is x. What is monthly revenue R=R(x)?
    (c) What rent per apartment will maximize monthly revenue R?

    I am quite stuck. I've tried writing out all the bits of data, but have no luck.

    I made a chart like this:

    Cost of 400 = 100 units rented = Revenue of 40000
    Cost of 410 = 98 units rented = Revenue of 40180
    Cost of 420 = 96 units rented = Revenue of 40320
    Cost of 430 = 94 units rented = Revenue of 41280
    Cost of 440 = 92 units rented = Revenue of 40,480
    Cost of 450 = 90 units rented = Revenue of 40,500

    I can see the revenue function will probably be quadratic. And I know the answer to (a), as you can see by the chart. But the rest has me quite frantic.

    Any help would be greatly appreciated.

    Thanks,

    - Jon Nash
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  2. #2
    MHF Contributor

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    The original rent is $400 and "for each $10 per month increase, there will be two vacancies with no possibility of filling them". If x is the monthly rent, then x- 400 is the amount of increase above $400 and (x- 400)/10 is the number of "$10 per month" increases. Since the rental of two apartments is lost for each such increase there will be 2(x-400)/10= (x- 400)/5 fewer apartments rented. That is, with rent at x, n(x)= 100- (x- 400)/5= 500/5- (x- 400)/5= (100- x)/5 apartments will be rented, each for x dollars per month. Now can you find R(x)? That will be a quadratic so you can find its vertex, answering (c), by completing the square.
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