Question 388399: I need help with the following quadratic equation:
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?
It would be nice to have this broken down so I can understand it.
Thank so much for your time. :) Answer by solver91311(18785) (Show Source):
It opens down because the lead coefficient is negative.
Ticket sales start small, get larger, then get smaller
Set the quadratic equal to zero. Use the quadratic formula to solve it. The integer part of the largest root is the last day of ticket sales.
Divide the opposite of the coefficient on the first degree term by 2 times the lead coefficient to get the value of x where the peak occurs.
Evaluate the function at the value of x determined above to find the number of tickets sold at the peak.
The x-coordinate of the vertex is the day when the peak occurs. The y-coordinate of the vertex is the number of tickets sold that day.
There are two solutions to the equation. I know because the value of the function at the vertex is positive and the parabola opens downward, therefore the graph must cross the x-axis in two places. I also know that there are two solutions because the lead coefficient and the constant coefficient have opposite signs and the quadratic is not a perfect square.
The solutions represent the days when there will be zero ticket sales. The negative root makes no sense because you can't be selling tickets before you start selling tickets.
My calculator said it, I believe it, that settles it