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Math Help - Help please....Optimization equation

  1. #1
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    Help please....Optimization equation

    I can't get the equation for this question, I spent almost 2 hours!! This is all new to me so sorry for any inconveniance.

    Here's the question:


    A Car rental agency has 24 identical cars. The owner of the agency finds that a price of $10 per day, all the cars can be rented. However for each $1 increase in rental, on of the cars is not rented. What should he carge to maximize income?


    Please help and thanks for even taking the time to help this noob.
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  2. #2
    TD!
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    If your price is $10, you rent all 24 cars; yielding: 10*24 = $240.
    If your price is $11, you rent only 23 cars; yielding: 11*23 = $253.
    If your price is $12, you're down to 22 cars; yielding: 12*22 = $264.

    You see the pattern? As you can see, the profit is rising.
    However, you can't keep doing this since the number of cars will go to zero.

    In general, if the price is (10+n), you'll be renting (24-n) cars.
    The income I(n) is still the product of the price with the number of cars.
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  3. #3
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    Thank you so much....we did an example similar to this in class, just a different variation which confused me a bit, hence the long hours of thinking. Anyway thanks a lot.
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  4. #4
    TD!
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    You're welcome. Did you manage to solve it, what do you get as solution?
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  5. #5
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    He needs to charge $17 to maximize income, and I also checked the second order condition to make sure its correct. Thanks again!
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  6. #6
    TD!
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    I found the same answer
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