Calculus Max Profit

To make pom poms in our school colors, we will have expenses of $100 to rent the Acme PomPom Plant, and then $0.25 per pom pom in materials. We believe that we can sell 500 pom poms if we charge $1.50. Assume the number sold is a linear function of the price. How much should we charge to maximize profit?

I have done this problem many times but I cannot find max profit. I keep fining a max income. I am lost and don't know what to do please help!

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  1. "Assume the number sold is a linear function of the price"
    but no information is given on the price elasticity, so we will assume, in general, an increase in price of $1 will increase sales by m units.
    Under normal supply-demand curve, m is necessarily negative, of the order of -200 or so.

    With that in mind, and knowing that (500,1.50) is a point on the line sales versus price, we construct the sales(y)-price(x) relation as:
    (y-500)=m(x-1.50)
    therefore, at a price of x, we expect sales of
    y=m(x-1.5)+500

    Total revenue,
    R=xy

    Total profit
    P=xy - cost
    =xy - (0.25x+100)
    =x(m(x-1.5)+500) - (0.25x+100)

    To get the maximum profit, we differentiate profit with respect to price, and equate to zero to find the optimum price:

    dp/dx = m*x+m*(x-1.5)+1999/4 =0
    Solve for x:
    x=(6*m-1999)/(8*m)

    If the price elasticity m=-100,
    x=2599/800=$3.25
    if m=-200
    x=3199/1600=$2
    if m=-300
    x=3799/2400=$1.58
    ...
    etc.

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