Problem of the Week
Problem B and Solution
Let’s Go to the Movies

Problem

At a movie theatre, all tickets are the same price. The ticket revenue is the money the theatre gets from customers when they buy tickets. The line graph shows the total ticket revenue when different amounts of tickets are sold.

1. What is the total ticket revenue when $100$ tickets are sold?

2. How much does one ticket cost?

3. The theatre has $250$ seats in total. What is the total ticket revenue if they sell out?

4. The theatre is planning an open air movie for which they will charge the same price per ticket. If the open air space can hold $600$ people, what is the maximum total ticket revenue for that show?

5. How can you tell from the graph that all tickets are the same price? Explain.

Solution

1. If $100$ tickets are sold, then we can use the graph to determine that the total ticket revenue will be $\mathrm{\$}1000$.

2. The cost for $1$ ticket is the same as the total ticket revenue for $1$ ticket. We can’t easily read the revenue from the graph when $1$ ticket is sold. However, we determined in (a) that the revenue for $100$ tickets is $\mathrm{\$}1000$. Since all tickets are the same price, then $1$ ticket costs $\mathrm{\$}1000÷100=\mathrm{\$}10$.

3. Since we determined in (b) that one ticket costs $\mathrm{\$}10$, then $250$ tickets cost $250\times \mathrm{\$}10=\mathrm{\$}2500$. So the total ticket revenue will be $\mathrm{\$}2500$ if they sell out.

   Alternatively, we can extend the graph to $250$ tickets sold by adding one more gridline to the right. When we extend the diagonal line, we will find that we also need to add one more gridline to the top, and that when we reach $250$ tickets sold, the total ticket revenue will be $\mathrm{\$}2500$.

4. The total ticket revenue will be at its maximum when all $600$ tickets are sold. Since we determined in (b) that one ticket costs $\mathrm{\$}10$, then $600$ tickets cost $600\times \mathrm{\$}10=\mathrm{\$}6000$. So the maximum total ticket revenue is $\mathrm{\$}6000$.

   Alternatively, we can use the graph to determine that when $200$ tickets are sold, the total ticket revenue will be $\mathrm{\$}2000$. Since $600=3\times 200$, then the total ticket revenue when $600$ tickets are sold is $3\times \mathrm{\$}2000=\mathrm{\$}6000$.

5. The graph is a straight line, which means that as the number of tickets sold increases, the total ticket revenue increases at a constant rate. This means that each ticket must be the same price.