Question

This is due tonight! I need all the help I can get thank you! :)

Answer 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=-0.4x^2+7x+13

Answer 2

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?

Answer 3

Okay so to figure out the last day of sales, we need to figure out when the number of tickets remaining will be zero since the equation determines the amount of tickets left after x days. so we take our equation: tickets=-0.4x^2+7x+13 and set it equal to zero to find when our tickets will run out.

This is a bit tricky to solve by hand, so hopefully you are allowed to use a calculator. Put it into the y= part of your graphing calculator and you can solve for when the graph crosses the x-axis (when it hits this axis, y is 0 so thus tickets=0). With a little help from my TI-89, I found that at x=19.1933, tickets=0. So that tells us that after more than 19 days (so 20 days) all the tickets have been sold. So our answer is 20 days.

To figure out if it has a peak or a low, you have to look at the sign in front of the x^2. If it is positive, it tells you that the graph opens upward and it has a minimum. Since ours is negative, it tells us that the graph opens DOWNWARD and thus has a maximum. So our answer is that the ticket sales will peak.

To figure out where the peak is, we use the equation x=-b/2a where we get a and b from the general formula of a quadratic ax^2+bx+c. So for our purposes, we have b=7 and a=-.4. When we plug it in, we get x=(-7)/(2(-.4)) =8.75. The equation we used gives us the line of symmetry of the graph, which also tells us where the max or min is. So after 8.75 days, the peak will occur.

To figure out what this peak is, we simply plug 8.75 into x into the equation like so: -.4(8.75)^2+7(8.75)+13=43.625 tickets.

The vertex is the highest or lowest point of a quadratic function. In our case, we just found this point. Our highest value occurs at (8.75,43.625). This relates to our previous answers because the 8.75 comes from our answer before when we were figuring out the days of the max and the 43.625 comes from that maximum value.

Regarding the number of solutions, we know that when the graph hits 0 at roughly 19.19 days this is a solution for our function. We know that since this is a quadratic graph, it should have two solutions, however it is possible that there are duplicates. So once again we solve for -.4x^2+7x+13=0 to get the other solution. In doing so, we get roughly -1.693.

The solutions represent where the number of tickets equal 0. The solution that does not make sense is the negative solution (-1.693) since we cannot express time, in our case days, as negative items since that would technically be traveling back in time!

Phew that was a lot. Hope it helped! :)