Investigation 6.2. Revenue from Theater Tickets.
The local theater group sold tickets to its opening night performance for \(\$5\) and drew an audience of \(100\) people. The next night, the group reduced the ticket price by \(\$0.25\) and \(10\) more people attended; that is, \(110\) people bought tickets at \(\$4.75\) apiece. In fact, for each \(\$0.25\) reduction in ticket price, \(10\) additional tickets can be sold.

Complete the table.
Number
of price
reductionsPrice of
ticketNumber
of tickets
soldTotal
revenue\(0\) \(5.00\) \(100\) \(500\) \(1\) \(4.75\) \(110\) \(522.50\) \(2\) \(\) \(\) \(\) \(3\) \(\) \(\) \(\) \(4\) \(\) \(\) \(\) \(5\) \(\) \(\) \(\) \(6\) \(\) \(\) \(\) \(7\) \(\) \(\) \(\) \(8\) \(\) \(\) \(\) \(9\) \(\) \(\) \(\) \(10\) \(\) \(\) \(\) \(11\) \(\) \(\) \(\) \(12\) \(\) \(\) \(\) 
Use your table to make a graph. Plot total revenue on the vertical axis versus number of price reductions on the horizontal axis.

Let \(x\) represent the number of price reductions, as in the first column of the table. Write algebraic expressions in terms of \(x\) for each quantity.The price of a ticket after \(x\) price reductions:\begin{equation*} \text{Price}= \end{equation*}The number of tickets sold at that price:\begin{equation*} \text{Number}= \end{equation*}The total revenue from ticket sales:\begin{equation*} \text{Revenue}= \end{equation*}
 Enter your expressions for the price of a ticket, the number of tickets sold, and the total revenue into the calculator as \(Y_1\text{,}\) \(Y_2\text{,}\) and \(Y_3\text{.}\) Use the Table feature to verify that your algebraic expressions agree with your table from part (1).
 Use your calculator to graph your expression for total revenue in terms of \(x\text{.}\) Use your table to choose appropriate window settings that show the high point of the graph and both xintercepts.
 What is the maximum revenue possible from ticket sales? What price should the theater group charge for a ticket to generate that revenue? How many tickets will the group sell at that price?