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23 July, 14:17

A car rental agency rents 190 cars per day at a rate of 29 dollars per day. for each 1 dollar increase in the daily rate, 5 fewer cars are rented. at what rate should the cars be rented to produce the maximum income, and what is the maximum income?

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  1. 23 July, 14:55
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    The equation that we can create from this situation is:

    i = (190 - 5 x) * (29 + x)

    where i is the income and x is the increase in daily rate

    Expanding the equation:

    i = 5510 + 190x - 145x - 5x^2

    i = - 5x^2 + 45x + 5510

    Taking the 1st derivative:

    di/dx = - 10x + 45

    Set to zero to get the maxima:

    -10x + 45 = 0

    x = 4.5

    So the cars should be rented at:

    29 + x = 33.5 dollars per day

    The maximum income is:

    i = (190 - 5*4.5) * (33.5)

    i = 5,611.25 dollars
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