suppose you are an event corrdinator 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 tickcets sell out. You provided the manger 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 thies 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 sole on the day when the peak or low occures?
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 -.02x^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?
a. To determine whether the graph of the equation opens up or down, we can look at the coefficient of the x^2 term. In this case, the coefficient is -0.2. Since the coefficient is negative, the graph of the equation will open downwards.
b. As time passes, the ticket sales will initially increase, reach a peak, and then start to decrease. This is because the quadratic equation represents a downward-opening parabola, which means the ticket sales will eventually decline.
c. To determine the last day that tickets will be sold, we need to find when the ticket sales reach zero. So, we can set the equation equal to zero:
-0.2x^2 + 12x + 11 = 0
By solving this quadratic equation, we can determine the last day tickets will be sold.
d. The tickets will peak during the middle of the sale. This can be determined by the fact that the graph of the quadratic equation is a downward-opening parabola, which means the highest point (peak) will occur at the vertex.
e. To find the number of days it takes to reach the peak, we can find the x-coordinate of the vertex. We can use the formula x = -b / (2a), where a and b are coefficients of the quadratic equation (-0.2x^2 + 12x + 11). The x-coordinate of the vertex represents the number of days.
f. To find the number of tickets sold on the day of the peak, we can substitute the number of days (found in part e) into the quadratic equation and solve for tickets.
g. The point of the vertex represents the maximum or minimum point of the quadratic equation. In this case, since the graph is opening downwards, the vertex represents the maximum point. The vertex includes the x and y coordinates, and the x-coordinate represents the number of days (related to parts e and f) while the y-coordinate represents the number of tickets sold on that day.
h. To determine the number of solutions to the equation -0.2x^2 + 12x + 11 = 0, we can use the discriminant. The discriminant is given by b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. Depending on the value of the discriminant, we can decide the number of solutions.
i. The solutions to the quadratic equation represent the days when the ticket sales reach zero. However, it is possible that one or both of the solutions won't make sense in the context of the problem. For example, if one of the solutions is negative or beyond the timeframe of ticket sales, it would not be meaningful in the given context.