1. Suppose you are an event coordinator for a large performance theater. One of the hottest new Broadway musicals has started to tour and your city is the first stop on the tour. You need to supply information about projected ticket sales to the box office manager. The box office manager uses this information to anticipate staffing needs until the tickets sell out. You provide the manager with a quadratic equation that models the expected number of ticket sales for each day x. ( is the day tickets go on sale).
Tickets = -0.2x^2 + 14x + 11
What is your question?
a. Does the graph of this equation open up or down? How did you determine this?
b. Describe what happens to the tickets sales as time passes.
c. Use the quadratic equation to determine the last day that tickets will be sold.
Note. Write your answer in terms of the number of days after ticket sales begin.
d. Will tickets peak or be at a low during the middle of the sale? How do you know?
e. After how many days will the peak or low occur?
f. How many tickets will be sold on the day when the peak or low occurs?
g. What is the point of the vertex? How does this number relate to your answers in parts e. and f?
h. How many solutions are there to the equation ? How do you know?
i. What do the solutions represent? Is there a solution that does not make sense? If so, in what ways does the solution not make sense?
To find the projected ticket sales for each day x, you can use the quadratic equation provided: Tickets = -0.2x^2 + 14x + 11.
Let's break down the equation:
-0.2x^2 represents the decreasing trend of ticket sales as the number of days since tickets went on sale increases. The coefficient -0.2 indicates that the ticket sales decrease at a certain rate as time passes.
14x represents the increasing trend of ticket sales as more days pass since tickets went on sale. The coefficient 14 indicates that ticket sales increase linearly with time.
11 represents an initial ticket sale value or a constant adding a baseline number of ticket sales independent of the number of days since tickets went on sale.
To calculate the projected ticket sales for a specific day, substitute the day's value for x into the equation. For example, if you want to find the projected ticket sales on day 5, substitute x = 5 into the equation.
Tickets = -0.2(5)^2 + 14(5) + 11
Tickets = -0.2(25) + 70 + 11
Tickets = -5 + 70 + 11
Tickets = 76
The projected ticket sales for day 5 would be 76.
You can repeat this process for any other day to find the projected ticket sales.