91Ó°ÊÓ

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 \(1981-2000 .\) (Source: The League of American Theaters and Producers, Inc.) (a) Use this model to estimate the attendance in the year \(1995 .\) Compare it to the actual value of 9 million. (b) Use this model to predict the attendance for the year 2006 (c) What is the vertex of the parabola associated with the function \(p\), and what does it signify in relation to this problem? (d) Would this model be suitable for predicting the attendance at Broadway shows for the year \(2025 ?\) Why or why not? (e) \(\quad\) Use a graphing utility to graph the function \(p\) What is an appropriate range of values for \(t ?\)

Step by step solution

01

Finding Attendance in 1995

To find the attendance in year 1995, we need to plug it into our function p. To do so, we subtract 1981 from 1995, since our function is based on years since 1981, to get \( t = 14 \). Now plug \( t \) in the function to get \( p(14) \). So, 1995's attendance would be approximately \( p(t) = 0.0489*14^{2} - 0.7815*14 + 10.31 \).

02

Predicting Attendance in 2006

To predict the attendance in year 2006 we do as in Step 1. Subtract 1981 from 2006 to get \( t = 25 \). And then substitute into the function to find \( p(25) = 0.0489*25^{2} - 0.7815*25 + 10.31 \).

03

Finding the Vertex of the Parabola

The vertex of a quadratic function is given by \( -b / (2*a) \) and \( f(-b / (2*a)) \) where in this case \( a = 0.0489 \) and \( b = -0.7815 \). We first find \( t \) by substituting \( a \) and \( b \) in the first formaula and then substitute \( t \) in \( p(t) \) to get the vertex of the parabola.

04

Predicting Suitable for 2025

The function is based on data from 1981 to 2000, so forecasting for 2025 might not be accurate as the data the prediction is based upon doesn't extend that far. That's almost 25 years past the latest year used to formulate the model. Traditionally, such extrapolations should be done with extreme caution and an understanding of the inherent limitations.

05

Use a Graphing Utility to Graph the Function p and Find Appropriate Range Values for t

Using a graphing utility like Desmos or a Graphing Calculator the function \( p(t) = 0.0489 t^{2} -0.7815 t + 10.31 \) can be put into the utility to get a visual representation of the function. The range for t should logically be between 0 and the present year (when the model was created) minus 1981, as those are the years the model is intended to cover.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabolic Models

Quadratic functions, like the one used to model Broadway show attendance, often create a parabolic shape when graphed. This model is essentially a U-shaped curve used to represent various real-world situations. In our case, the parabola helps us understand how attendance has changed over the years and possibly predict future trends.

Parabolic models are useful because they can simulate growth, decline, or stability depending on real-world data. The equation given, \( p(t) = 0.0489 t^2 - 0.7815 t + 10.31 \), shows this phenomenon by describing attendance in terms of years since 1981. Given parameters are crucial:

• The coefficient \(0.0489\) suggests how rapidly attendance's growth or decline might be.
• The middle term \(-0.7815 t\) can indicate an initial decline trend.
• You find specific attendance numbers by substituting \(t\) with the desired year calculation (e.g., \( t = 14 \) for 1995).

By understanding the composition of these models, it's easier to comprehend how certain trends in data are represented over time.

Vertex of a Parabola

The vertex of a parabola is a significant point that can offer insights into the peaking or lowest value of a quadratic graph.

To find the vertex of the quadratic function \( p(t) = 0.0489 t^2 - 0.7815 t + 10.31 \), we use the formula \( t = -\frac{b}{2a} \). For our function:

• Using \( a = 0.0489 \) and \( b = -0.7815 \), plug into the formula to get \( t = -\frac{-0.7815}{2*0.0489} \).
• This gives us the year since 1981 when attendance peaked.
• The vertex point also helps determine how attendance behaves around this peak point.

When interpreting this in relation to Broadway shows, the vertex tells us the maximum or minimum attendance that could have occurred during the years data were collated.

So, understanding the vertex's significance can guide us in making informed predictions about attendance trends.

Predictive Modeling

Predictive modeling using quadratic functions relies heavily on historical data to estimate future outcomes. The function \( p(t) = 0.0489 t^2 - 0.7815 t + 10.31 \) is derived from data between 1981 and 2000.

While the model can be used to predict attendance for a few years beyond 2000 (like 2006), its reliability diminishes the farther we go into the future, especially for a year like 2025.

• The dataset doesn’t cover trends that may have emerged after 2000.
• Extrapolations beyond the data range should be done with caution, recognizing limitations and potential inaccuracies.
• External factors impacting attendance, not accounted for in the model, could significantly alter predictions.

Predictive modeling is powerful, but it requires thoughtful consideration of the data range and understanding of what the model truly holds within its scope. By grasping these concepts, you can better judge when predictions are trustworthy or when other data considerations might be needed.