SOLUTION: a hall charges $30 per person for a sports banquet when 200 people attend. For every 10 extra people that attend, the hall will decrease the price by $1.50per person. What number o

Question 390129: a hall charges $30 per person for a sports banquet when 200 people attend. For every 10 extra people that attend, the hall will decrease the price by $1.50per person. What number of people will maximize the revenue of the hall?
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
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 (floor represents the floor value function) Since x people attend, the profit P is

Suppose that x is a multiple of 10, that is, for nonnegative integer y. Then,

. 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.
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