calculus

Maximimum profit:
you operate a tour service that offers the following rates: (1) $200 per person if 50 people (the minimum number to book the tour) go on the tour. (2) for each additional person, up to the maximum of 80 people total, everyones charge is reduced by $2. it costs you $6000 (a fixed cost) plus 32 per person to conduct the tour. how many people does it take to maximize your profit? (hint: you will need to understand the words revenue, cost, and profit.)

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  1. My applied calculus is a little rusty but I think I figured it out.

    let x=# of people over 50 that attend

    your revenue function would be:
    (50+x)(200-2x)
    10000+100x-2x^2

    Cost function:
    6000+32(50+x)
    7600+32x

    Profit function (subtract the two):
    Yp=(-2x^2+100x+10000)-(7600+32x)
    simplify
    Yp=-2x^2+68x+2400

    Take derivative
    y'=-4x+68

    Set equal to zero and solve
    0=-4x+68
    4x=68
    x=17

    But remember we let x equal to number of attendees over 50, so we have to add it to get our total number of people
    67 people will maximize your profit

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