Question

3000-Meter Speed Skating The winning times for the women's 3000 -meter speed skating race at the Olympics from 1960 to 2006 can be approximated by the quadratic model \(y=0.0202 t^{2}-5.066 t+550.24, \quad 60 \leq t \leq 106\) where \(y\) is the winning time (in seconds) and \(t\) represents the year, with \(t=60\) corresponding to \(1960 .\) (Sources: World Almanac and Book of Facts 2005 and NBC) (a) Use a graphing utility to graph the model. (b) Use the model to predict the winning times in 2010 and \(2014 .\) (c) Does this model have a horizontal asymptote? Do you think that a model for this type of data should have a horizontal asymptote?

Short answer

A graph of the model is first drawn. Next, the equation is used to predict winning times for 2010 and 2014, which results in the time for 2010 as approximately 243 seconds, and for 2014 as approximately 244 seconds, assuming no rounding errors. The model does not have a horizontal asymptote because quadratic functions don't. As for whether a model for this kind of data should have an asymptote, it can be argued that there could be a physical limit to how fast a human can skate, suggesting the usefulness of a horizontal asymptote in future models, though it has not yet been formally established.

Step-by-step solution

Graphing the Model

Use a graphing utility to plot the quadratic model \(y=0.0202 t^{2}-5.066 t+550.24, \quad 60 \leq t \leq 106\), where \(y\) represents winning time and \(t\) represents the year.

Predicting Winning Times

To predict winning times in 2010 and 2014, use \(t = 150\) (for 2010) and \(t = 154\) (for 2014) and substitute these into the model. For 2010: \(y= 0.0202*(150)^2 - 5.066*150 + 550.24\) and for 2014: \(y= 0.0202*(154)^2 - 5.066*154 + 550.24\).

Model Asymptote Analysis

A quadratic function does not have any horizontal asymptotes since the graph extends indefinitely upward or downward as \(x\) approaches positive or negative infinity. However, as to whether a model for this type of data should have a horizontal asymptote is subjective. It could be argued that there might be a limit beyond which human speed cannot improve, suggesting the usefulness of a horizontal asymptote in future modeling; yet, this limit has not been formally established.

Key concepts

Graphing Utility

A graphing utility is a powerful tool that allows you to visually represent mathematical models, such as the quadratic model given for the Olympic speed skating times. By inputting the equation \(y=0.0202 t^{2}-5.066 t+550.24\), you can plot the graph over the specified range \(60 \leq t \leq 106\). This visual representation helps you understand how the winning times have changed over the years and can be instrumental in analyzing trends in the data.
Graphing utilities, like graphing calculators or software, provide a simple and accurate way to see the shape of a function. They can plot points, lines, and curves, making the task of interpreting functions much more manageable. By examining the graph, you can quickly identify key characteristics of the model such as its vertex, axis of symmetry, and direction of opening (upwards or downwards in the case of quadratics).
When plotting the quadratic function, pay attention to the behavior of the parabola. The graph's curve will show past data approximations, and by extending it, you can make future predictions. Using a graphing utility brings the equation to life, allowing for an intuitive grasp of abstract mathematical concepts.

Predictions

Predictions involve using the mathematical model to foresee future outcomes. In the context of this exercise, we need to predict the winning times for the women's 3000-meter speed skating in the 2010 and 2014 Olympics using our quadratic equation.
To make these predictions, substitute the corresponding years into the model's formula. For 2010, use \(t = 150\) because 2010 is 150 years past 1860, and for 2014, use \(t = 154\).

• For 2010: Calculate \(y= 0.0202(150)^2 - 5.066(150) + 550.24\).
• For 2014: Calculate \(y= 0.0202(154)^2 - 5.066(154) + 550.24\).

The predictions you obtain will offer an approximation of the winning times. This is a valuable exercise as it applies the theoretical model to real-world data, showcasing the practicality of mathematical modeling. It also highlights how predictive models can assist in statistical forecasting, essential in many scientific and business fields.

Horizontal Asymptote

Understanding horizontal asymptotes can be tricky but it's important in analyzing certain types of functions. In the quadratic function given, there is no horizontal asymptote because as the value of \(t\) (or years) goes infinitely large, the quadratic expression grows without bounds.
An asymptote is a line that a graph approaches but never touches or crosses. For rational functions, horizontal asymptotes give information about the behavior of the function as it goes towards infinity. However, quadratic functions, by their nature, do not have this type of asymptote because they curve either upwards or downwards indefinitely as the inputs increase or decrease.
When considering whether a model should have a horizontal asymptote, think about the data's nature. For example, one might theorize there is a "limit" to how fast humans can skate, reflecting some horizontal leveling in future models. However, this limit is speculative and has not yet been explicitly determined in this context. The concept is thought-provoking, as it invites consideration of the ultimate constraints of human performance.

Olympic Times

Olympic times, especially in events like speed skating, serve as a fascinating dataset for mathematical modeling. The changing times reflect improvements in athletic performance, advancements in technology, and even changes in competition conditions.
The quadratic model used for Olympic times from 1960 to 2006 attempts to capture the trend in performance over time. It's typical to use mathematical models to analyze and understand past trends as they can provide insights into future performance.

• This model is particularly useful in observing how variables such as technology improvements and athlete training affect performance.
• Examining historical Olympic times can offer insights into the progress in various sports disciplines.

Mathematical modeling, like the one presented, aids in comprehending these trends. Such models not only reflect performances but also help set realistic goals and expectations for future Olympic events. This exercise demonstrates how sports statistics coupled with math can be both engaging and educational, shedding light on the ever-evolving landscape of athletic achievement.