Ticket Sales
Living in or near a metropolitan area has some advantages. Entertainment opportunities are almost endless in a major city. Events occur almost every night, from sporting events to the symphony. Tickets to these events are not available long and can often be modeled by quadratic equations.
Answer the following questions..
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).
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 answer these questions about the projected ticket sales, we'll use the quadratic equation that models the expected number of ticket sales for each day x. Let's denote the equation as y = ax^2 + bx + c, where x represents the number of days after ticket sales begin.
a. To determine whether the graph of the equation opens up or down, we need to look at the coefficient of the x^2 term (a). If a > 0, the graph opens up, and if a < 0, it opens down.
b. As time passes, the ticket sales will either increase or decrease based on the direction the graph opens. If the graph opens up, the ticket sales will increase as the days go by. If it opens down, the ticket sales will decrease.
c. To determine the last day that tickets will be sold, we need to find the x-coordinate of the vertex of the quadratic equation. The vertex is given by the formula x = -b/(2a). Calculate this value using the coefficients a and b from the quadratic equation.
d. Whether the tickets will peak or be at a low during the middle of the sale depends on the direction the graph opens. If the graph opens up, the tickets will be at a low in the middle. If the graph opens down, they will peak in the middle.
e. To find the number of days when the peak or low occurs, we need to find the x-coordinate of the vertex.
f. Once you have the x-coordinate of the vertex, substitute it into the equation to find the number of tickets sold on that day (y-value).
g. The point of the vertex represents the maximum or minimum point on the graph of the quadratic equation. If the graph opens up, the vertex represents the minimum (low) point, and if it opens down, it represents the maximum (peak) point. The number of tickets sold on the day of the vertex corresponds to the y-value of the vertex.
h. The number of solutions to the equation depends on the discriminant (∆ = b^2 - 4ac). If ∆ > 0, there are two distinct solutions, if ∆ = 0, there is one solution, and if ∆ < 0, there are no real solutions.
i. The solutions to the equation represent the days when ticket sales reach zero. In this context, they would represent the last day of ticket sales. If there is a solution that does not make sense, it could be due to a negative value for x, which would mean the event hasn't even started yet.