"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:
therefore, at a price of x, we expect sales of
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:
If the price elasticity m=-100,
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