Question 585577: Please help I am having real trouble with this. Thank you!!!!
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.
Application Practice
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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 = -x^2 + 5x + 29 = 0
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 -x^2 + 5x + 29 = 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 by KMST(5328)   (Show Source):
You can put this solution on YOUR website! a.  represents a parabola.
The sign of the coefficient of the term in  tells you if the graph opens up or down. You could memorize a rule, but you don't need to because math makes sense (but only if you don't panic).
You can see the value for x=0. It's y=29. And for x=1, y=-1+5+29=33.
Think of large values of x, and what happens to the y value as x gets larger and larger.
For x=10  .
For x=100  .
IT's obvious that as x grows larger and larger your graph goes down faster and faster, and will not come back up. It opens down.
b. It started going up from x=0 to x=2, but parabolas that open down go up, crest and then go down. It makes sense for sales of something to do the same (although an event that it is heavily advertised ahead of time may have top sales the first day).
c. The function does not make sense for x<0 (before ticket sales begin), or once y<0 (when a negative number of tickets would be sold. The model is supposed to be a good estimate of sales in between those two points.
 can be solved to find two solutions. One is negative and makes no sense. The other can be found using the quadratic formula as:
 or approximately 8.44.
So 8.44 days after ticket sales begin zero tickets would be sold, according to the model.
That means that we expect some sales 8 days after tickets go on sale, but no sales the next day.
d. Tickets sales will peak during the middle of the sale. We know because the coefficient of is  and that makes the graph open down. It was discussed above.
e. After how many days will the peak or low occur? The formula memorized says
 where a and b are coefficients, a=-1 and b=5 in this case.
So  -->  , but since  is between 2 and 3, the most ticket sales will occur 2 and 3 days after sales begin. As a parabola is symmetrical around the vertex (maximum in this case), and is at the same distance between 2 and 3, the same number of tickets are expected to be sold both days.
f. How many tickets will be sold on the day when the peak or low occurs?
For x=2 (and for x=3), substituting you find y=35.
g. What is the point of the vertex? How does this number relate to your answers in parts e. and f? See part e above.
h. How many solutions are there to the equation -x^2 + 5x + 29 = 0? How do you know? There are 2. When we solve the quadratic equation in part c, the discriminant is 141, the number under the square root. If it is positive there are 2 real solutions. If zero, there is one. If negative, there is no real solution.
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?
See part c above. A negative x does not make sense.
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