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.
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^2 + 12x + 11
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 -0.2x^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?
To answer these questions about the quadratic equation Tickets = -0.2x^2 + 12x + 11, let's break it down step by step:
a. To determine whether the graph of the equation opens up or down, you look at the coefficient of the x^2 term, which is -0.2. If this coefficient is negative, the graph will open downwards. In this case, since -0.2 is negative, the graph opens down.
b. As time passes (as the value of x increases), the number of ticket sales can be determined by plugging in different values for x into the quadratic equation. The result will give you the expected number of tickets sold for that day. The sales will fluctuate or vary based on the quadratic equation, which represents a curve.
c. To determine the last day that tickets will be sold, you need to find the x-value where the number of tickets sold is zero. In other words, you need to solve the equation -0.2x^2 + 12x + 11 = 0. This can be done by factoring, completing the square, or using the quadratic formula.
d. To determine whether the tickets will peak or be at a low during the middle of the sale, you can look at the coefficient of the quadratic term (-0.2x^2). If this coefficient is negative, the graph will have a peak, and if it's positive, the graph will have a low point. In this case, since the coefficient is negative, the graph will have a peak.
e. To find the number of days it takes for the peak to occur, you can use the formula x = -b / (2a), which gives you the x-coordinate of the vertex of the quadratic equation. In this equation, a = -0.2 and b = 12.
f. Once you find the number of days for the peak, you can plug that value into the quadratic equation to find the number of tickets sold on that day.
g. The point of the vertex is the highest or lowest point of the quadratic curve. It represents either the peak or low point of ticket sales, depending on whether the coefficient of the quadratic term is negative or positive. The x-coordinate of the vertex relates to the answer in part e, as it gives you the number of days it takes for the peak or low to occur. The y-coordinate of the vertex relates to the answer in part f, as it gives you the number of tickets sold on that day.
h. To determine the number of solutions to the equation -0.2x^2 + 12x + 11 = 0, you can use the discriminant of the quadratic equation. The discriminant is b^2 - 4ac, where a = -0.2, b = 12, and c = 11. If the discriminant is positive, there are two real solutions. If it's zero, there is one real solution. If it's negative, there are no real solutions. You can calculate the discriminant using these values.
i. The solutions to the quadratic equation represent the values of x at which the number of tickets sold is zero or the event is over. In the context of this problem, the solutions represent the number of days after ticket sales begin. It's possible to have a solution that does not make sense if it's a negative value or a fraction, as you can't have a negative number of days or a fraction of a day.