Problem 43 - Precalculus : Building Concepts and Connections

Question

Attendance at Broadway shows in New York can be modeled by the quadratic function \(p(t)=0.0489 t^{2}-0.7815 t+10.31,\) where \(t\) is the number of years since 1981 and \(p(t)\) is the attendance in millions. The model is based on data for the years \(1981-2000 .\) For which years was the attendance above 8 million? (Source: The League of American Theaters and Producers, Inc.)

Step-by-Step Solution

Answer

The attendance at the Broadway shows was above 8 million in the years between \(t_{1}+1981\) and \(t_{2}+1981\), where \(t_{1}\) and \(t_{2}\) are solutions for the given quadratic function.

Step 1: Find the Roots of the Quadratic Function

First, set the function equal to 8 to find the roots of the quadratic function, which are the values of t where the attendance is exactly 8 million. Solve the equation \(0.0489 t^{2}-0.7815 t+10.31 = 8\)

Step 2: Solving the Quadratic Equation

Solving the quadratic equation can be done by using the quadratic formula, which is \(t = [-(-0.7815) \pm \sqrt{(-0.7815)^{2}-4*(0.0489)*(10.31-8)}] / 2*(0.0489)\)

Step 3: Finding the Range of Years

We'll get two solutions for t, let's call them \(t_{1}\) and \(t_{2}\) with \(t_{1} \leq t_{2}\). The attendance is above 8 million for the years after 1981, between \(t_{1}\) and \(t_{2}\), because a quadratic function is a parabola and it's above y=8 between these two roots.

Key Concepts

Modeling Real-World Data

In many situations, including attendance at Broadway shows, it's important to represent complex, real-world data with mathematical models. Quadratic functions are one of the effective tools for this because they can model a variety of patterns, especially those that peak or have parabolic distributions.
Quadratic functions have the general form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. They are particularly useful for modeling situations where there's both a steady change over time and a turning point, such as maximum attendance in our Broadway example.
When a quadratic function is used, it allows predictions about one variable based on another. In the context of the Broadway attendance, the quadratic model \( p(t) = 0.0489t^2 - 0.7815t + 10.31 \) describes how attendance has changed over time, with \( t \) representing the years since 1981. This function can be used to both understand past behavior and make future predictions based on the modeled trend.

Solving Quadratic Equations

Solving quadratic equations is a key process when working with quadratic functions. It involves finding the values of \( t \) that make the equation true. Here, we equate the function to 8 million to find when attendance was above that level: \( 0.0489t^2 - 0.7815t + 10.31 = 8 \).
Solving involves rearranging the equation to the standard quadratic form \( ax^2 + bx + c = 0 \), resulting in \( 0.0489t^2 - 0.7815t + 2.31 = 0 \). We then apply different methods like factoring, completing the square, or using the quadratic formula to find the "roots" or solutions for \( t \).
In our example, we use the quadratic formula because the coefficients are not easily factored. This process finds two \( t \) values that represent moments when attendance hit exactly 8 million, which helps in understanding the periods of time attendance exceeded this number.

Using the Quadratic Formula

The quadratic formula is a universal method to solve any quadratic equation, given by \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It's a powerful tool because it directly gives the solutions without needing trial and error. In our case, \( a = 0.0489 \), \( b = -0.7815 \), and \( c = 2.31 \).
Applying these values into the formula gives two results called \( t_1 \) and \( t_2 \). These results indicate when attendance precisely equals 8 million, essentially giving the boundaries for when the attendance went above 8 million.
The discriminant, \( b^2 - 4ac \), within the formula indicates the nature of the roots – if it's positive, we get two distinct real roots as expected in this scenario. Understanding the quadratic formula not only provides solutions but also insights into the nature of the quadratic equations, their graphs, and how they relate to real-world data.