Investigation 4.2.1. 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 they 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.
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Fill in the table
No. of price reductions Price of ticket No. of tickets sold Total revenue \(0\) \(5.00\) \(100\) \(500.00\) \(1\) \(4.75\) \(110\) \(522.50\) \(2\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(3\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(4\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(5\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(6\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(8\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(10\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) -
On the grid below, plot Total revenue on the vertical axis versus Number of price reductions on the horizontal axis. Use the data from your table.
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Let \(x\) represent the number of price reductions, as in the first column of your table. Write algebraic expressions in terms of \(x\) forThe 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 \(Y_1, ~Y_2,\) 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 \(x\)-intercepts.
- 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?