Problem 85 - 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 of dollars. The model is based on data for the years \(1981-2000 .\) When did the attendance reach \(\$ 12\) million? (Source: The League of American Theaters and Producers, Inc.)

Step-by-Step Solution

Answer

The attendance reached $12 million approximately 15 years after 1981.

Step 1: Formulate the equation

Set the equation \( p(t) = 12 \). Substitute the value of \( p(t) \) in the given equation, which gives us \( 0.0489t^{2} - 0.7815t + 10.31 = 12 \).

Step 2: Simplify the equation

Subtract 12 from both sides of equation to bring it into standard quadratic form. This yields: \( 0.0489t^{2} - 0.7815t - 1.69 = 0 \).

Step 3: Solve for \( t \)

Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) to solve for \( t \). Here, \( a = 0.0489, b = -0.7815 \) and \( c = -1.69 \). Solving yields two solutions.

Step 4: Interpret the solution

Since \( t \) signifies years since 1981, only the positive solution is meaningful in this context. Thus, omit the negative solution.

Key Concepts

Attendance Modeling

Attendance modeling is an invaluable tool for understanding trends and changes over time. In the exercise, attendance at Broadway shows is modeled using a quadratic function. This function describes how attendance has evolved from 1981 to 2000. The variable \( t \) in the function represents years since 1981, indicating a growing or declining trend in attendance over those years.

By using this mathematical model, predictions can be made about attendance levels for specific years, helping event organizers and stakeholders make informed decisions. For example, in our task, we needed to determine when attendance reached $12 million by finding the value of \( t \) that satisfies the quadratic equation. Such models are crucial as they offer a macro perspective on data, making it easier to forecast future trends and prepare financially and logistically.

Quadratic Formula

The quadratic formula is a powerful method for solving quadratic equations, which are equations that can be expressed in the form \( ax^2 + bx + c = 0 \). This formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). In our Broadway attendance problem, we used specific values for \( a \), \( b \), and \( c \) derived from modifying the original equation to match this standard quadratic form.

Using the quadratic formula, we can determine the solutions for \( t \). These solutions tell us the possible years in which attendance hit $12 million. The formula is remarkable because it provides a direct solution method for any quadratic equation, ensuring accuracy and efficiency in problem-solving.

• Make sure to carefully calculate the discriminant \( b^2 - 4ac \). It tells us how many real solutions there are.
• Under the square root, a negative discriminant means no real solutions exist, while a zero or positive discriminant results in one or two real solutions, respectively.

Mathematical Modeling

Mathematical modeling involves creating equations or formulas to simulate real-world scenarios. In this exercise, the quadratic model serves to replicate the attendance trends of Broadway shows. This process involves analyzing historical data and representing it in a mathematical framework.

A crucial part of modeling is decision-making based on data. When a model is accurately constructed, it can offer insights that aren't immediately obvious. For instance, in our scenario, the historical data from 1981 to 2000 helped build our quadratic equation, enabling predictions about attendance in future or past years.

• Accurate data collection ensures model reliability, thus better predictions.
• Understand assumption limitations in models as they guide the interpretation of findings.

Mathematical modeling isn’t just about numbers; it’s a tool for enhancing understanding and strategic planning.

Problem Solving Steps

Problem-solving in mathematics often follows structured steps to ensure comprehension and correctness. In our exercise, the steps were systematically laid out to achieve the solution.

The first step involved identifying and correctly formulating the problem — setting \( p(t) = 12 \). We then progressed to manipulating the initial equation to isolate the quadratic expression. This was essential to use the quadratic formula effectively. Solving the quadratic equation using the formula then followed, which involved careful application of the formula to derive the correct value(s) for \( t \).

Lastly, interpretation of the mathematical solution within the real-world context was crucial. While solving provides multiple solutions, understanding which solution is meaningful for the problem is key. In this context, negative values for \( t \) had no practical implication and were thus discarded.

• Identify the problem and set an equation matching the scenario.
• Manipulate equations into useful forms, such as standard quadratic equations.
• Solve, then evaluate the results in context for real-world relevance.

Following these steps not only aids in arriving at the correct answer but ensures a clear understanding of the process.