Expert: Abe Mantell - 6/10/2010

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^+14+5(this is the governing equation that is referred to in part c below)

  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. Solve the governing equation using the quadratic formula.
  
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^+14x+5=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
I think you meant: "-0.2x^2 + 14x + 5" -- Yes?  If so, then...

a) down, since the coefficient of x^2 is negative

b) sales increases until x=35, then decreases

c) Do you mean solve for when ticket sales = 0, yes?

d) Maximum occurs when x=35, which gives a maximum sales of 250

e) 35 days [x=-b/(2a)=-14/(2(-0.2))]

f) 250, let x=35 in the equation

g) (35, 250) x=35 & T=250

h) There are 2 mathematical solutions, but one is negative, which
.  does not make sense for the problem.  So only one practical answer.
.  There have to be 2 solutions, since it is an "upside down" parabola
.  that is shifted up...so the two branches that point down must intersect
.  the x-axis.

i) see (h)

Abe