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16 October, 06:03

A health club currently charges its 1500 clients monthly membership dues of $ 44. The board of directors decides to increase the monthly membership dues. Market research shows that each $ 1 increase in dues will result in the loss of 5 clients. How much should the club charge each month to optimize the revenue from monthly dues?

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  1. 16 October, 06:10
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    The club should charge $172 to optimize the revenue.

    Step-by-step explanation:

    Hi there!

    The revenue can be calculated as the number of clients times the value of the membership dues. Let y be the revenue, then:

    y = 1500 · 44

    Let x be the increase in membership dues, then for a increse of x, the number of clients lost will be 5x:

    y = (1500 - 5x) (44 + x)

    y = 66000 - 220x + 1500x - 5x²

    y = - 5x² + 1280x + 66000

    It is a quadratic equation, a negative parabola. The maximum is the vertex of the parabola that is calculated as:

    x = - b/2a

    Where:

    b = 1280

    a = - 5

    Then:

    x = - 1280/2 · (-5)

    x = 128

    The club should charge 44 + 128 = $172 to optimize the revenue.
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