Expert: Scott A Wilson - 10/16/2009

Question
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. (x = 1 is the day tickets go on sale).   
         
Tickets = -0.2x^2 + 12x +11

  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 -0.2x^2 + 12x +11 = 0? 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?  

Answer
Note: if I get too busy on some day, I wouldn't answer this.  However, I have less than half a dozen questions pending.

The quadratic equation is the solution to y = ax² + bx + c.
It is y = (-b±√(b²-4ac)/(2a).
The ± sign indicates that there are two choices, (-b+√(b²-4ac)/(2a) and (-b-√(b²-4ac)/(2a).

Here, we are given -0.2x^2 + 12x +11.  That is a parabola with a = -0.2, b = 12, and c = 11.
The solutions would be (-12±√(12²-4(-0.2)11))/(2(-0.2)).
Now 12² = 144, 4(-0.2)11 = -8.8, and 2(-0.2) = -0.4.
This is then (-12±√(144+8.8))(-0.4).  -0.4 = -2/5, so mulitiply by -5/2.
That gives (60±5√152.8)/2.

a. Does the graph of this equation open up or down? How did you determine this?
Parabolas always open down if a is negative.

b. Describe what happens to the tickets sales as time passes?
They would rise ever slower until 30 days had passed.

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.)
The last ticket would be sold on the 30th day.

d. Will tickets peak or be at a low during the middle of the sale? How do you know?
They will be at a high since we have a parabola with a peak in the middle.
They are governed by a parabola that opens down.

e. After how many days will the peak or low occur?
The peak will be after 30 days.  The low will be on day 60.

f. How many tickets will be sold on the day when the peak or low occurs?
The peak will be 191 and the end will be 11.

g. What is the point of the vertex?
How does this number relate to your answers in parts e and f?
The vertex is the midpoint.  Since it opens down, it is the max.
It gives you the day with the most sold.

h. How many solutions are there to the equation -0.2x^2 + 12x +11 = 0? How do you know?
There are only two, since a parabola only crosses the x-axis in two places.

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?
The solutions indicate the day when no tickets will be sold.  Since the value is negative on the 61st day, this indicates sales will stop on the 60th day.  The first solution is less than 0,
but this makes no sense since that is the number of tickets sold before the play even starts selling tickets.