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 opens downward.
b. As time passes, the ticket sales will initially increase, reach a peak, and then start decreasing.
c. To determine the last day that tickets will be sold, we need to find the value of x when the ticket sales reach zero. We can set the equation equal to zero: -0.2x^2 + 12x + 11 = 0. Then, we can find the solutions to this quadratic equation.
d. Tickets will peak during the middle of the sale because the graph opens downward and the coefficient of the x^2 term is negative. This indicates that the highest ticket sales will occur at the vertex.
e. To find the number of days it will take for the peak or low to occur, we need to find the x-coordinate of the vertex. We can use the formula x = -b / (2a), where a and b are the coefficients of the quadratic equation (-0.2x^2 + 12x + 11).
f. Once we have determined the number of days for the peak or low to occur (the x-coordinate of the vertex), we can substitute this value into the quadratic equation to find the number of tickets sold on that day.
g. The point of the vertex represents the maximum or minimum point on the graph of the quadratic equation. The x-coordinate of the vertex obtained in part e represents the number of days it takes to reach the peak or low in ticket sales. The y-coordinate of the vertex represents the maximum or minimum number of tickets sold on that day.
h. To determine the number of solutions to the quadratic 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. If the discriminant is greater than zero, there are two real solutions. If the discriminant is zero, there is one real solution. If the discriminant is less than zero, there are no real solutions.
i. The solutions to the quadratic equation represent the values of x when the ticket sales are equal to zero. In this context, it means the days when all the tickets are sold out. There may be a solution that does not make sense if it falls outside the expected range of ticket sales or beyond the available time period for ticket sales.