Question 390129
We have to take into account that if 208 or 209 people attend, then the price is still $30 per person, but if 210 people attend, the price is $28.50.


Suppose x is the number of people that attend. Then, the price for each person is {{{30 - (1.50)(floor((x-200)/10))}}} (floor represents the floor value function) Since x people attend, the profit P is


{{{P = 30x - 1.50x*(floor((x-200)/10))}}}


Suppose that x is a multiple of 10, that is, {{{x = 200 + 10y}}} for nonnegative integer y. Then,


{{{P = 30(200 + 10y) - 1.50(200 + 10y)(y) = 600 + 300y - 300y - 15y^2 = 600 - 15y^2}}}. It is seen that P is maximized when y = 0, and that y = 0 produces a greater profit than y = 1.


Thus, the number of people that will maximize profit is between 200 and 209, inclusive (since 210 decreases sales per person and also profit). However, since each person pays the same amount in any case, the maximum profit occurs when 209 people attend.