We first find the function of total profit if tickets cost dollars each. To find the profit, we use the formula
We know that the cost of one ticket is , so all we need to do is find a formula for the total number of tickets sold based on the price of one ticket. This is given to us in the problem: at , the number purchased is , but for every additional dollar after , this total drops by . In symbols,
Now, all that remains is to find the critical points of in order to determine the maximum of .