Problem 79
Question
100-Meter Freestyle The winning times for the men's 100-meter freestyle swim at the Olympics from 1952 to 2004 can be approximated by the quadratic model \(y=86.24-0.752 t+0.0037 t^{2}, \quad 52 \leq t \leq 104\) where \(y\) is the winning time (in seconds) and \(t\) represents the year, with \(t=52\) corresponding to \(1952 .\) (Sources: The World Almanac and Book of Facts 2005 ) (a) Use a graphing utility to graph the model. (b) Use the model to predict the winning times in 2008 and \(2012 .\) (c) Does this model have a horizontal asymptote? Do you think that a model for this type of data should have a horizontal asymptote?
Step-by-Step Solution
Verified Answer
The graph of the model should be a downward opening parabola. Predicted winning times for 2008 and 2012 can be estimated by substituting respective \(t\) values into the equation. This quadratic model does not have a horizontal asymptote. However, in reality, the winning times cannot indefinitely decrease hence implying that the model for this type of data should ideally have a horizontal asymptote.
1Step 1: Graph the Model
To graph the model, you will need a graphing utility. Input the given quadratic function \(y=86.24-0.752t+0.0037t^{2}\) into the graphing utility. Remember to set the range of \(t\) from 52 to 104, which represents the years from 1952 to 2004.
2Step 2: Predict the Winning Times
To predict the winning times for 2008 and 2012, substitute the corresponding \(t\) values into the model. For 2008, \(t= 2008-1900=108\). For 2012, \(t= 2012-1900=112\). Solve the quadratic equations to find the respective \(y\) values, which represent the winning times.
3Step 3: Evaluate for Horizontal Asymptote
A quadratic function does not have a horizontal asymptote. However, data of winning sprint times in reality might approach a lower limit due to human physical constraints. An ideal model would therefore often include a horizontal asymptote reflecting this limit.
Key Concepts
Graphing Utilities in AlgebraPredicting with Quadratic EquationsHorizontal Asymptote Analysis
Graphing Utilities in Algebra
Graphing utilities, like graphing calculators or computer software, are powerful algebraic tools that help visualize equations and their solutions. For example, with the quadratic model of the Olympic 100-meter freestyle winning times, plotting the function \(y=86.24-0.752t+0.0037t^{2}\) allows us to see the curve and how the times change with each Olympic year.
In algebra, graphing is an essential aspect, especially when dealing with polynomial functions. It helps to identify the function's shape, intercepts, and direction of opening. In this case, the year \(t\) is plotted on the x-axis and the winning time \(y\) on the y-axis. The range is specified from \(t=52\) to \(t=104\) to match the years of interest, from 1952 to 2004. Through graphing, we interpret the model's behavior within context, a skill invaluable for both student learning and real-world application.
When graphing, it is important to set an appropriate scale and window size to accurately represent the data. The curve of the quadratic function will show the general declining trend in winning times, indicating improvements in athletic performance over the years.
In algebra, graphing is an essential aspect, especially when dealing with polynomial functions. It helps to identify the function's shape, intercepts, and direction of opening. In this case, the year \(t\) is plotted on the x-axis and the winning time \(y\) on the y-axis. The range is specified from \(t=52\) to \(t=104\) to match the years of interest, from 1952 to 2004. Through graphing, we interpret the model's behavior within context, a skill invaluable for both student learning and real-world application.
When graphing, it is important to set an appropriate scale and window size to accurately represent the data. The curve of the quadratic function will show the general declining trend in winning times, indicating improvements in athletic performance over the years.
Predicting with Quadratic Equations
Predicting outcomes with quadratic equations is a common practice in various fields, including sports analytics, economics, and engineering. Quadratic equations often describe trends that involve acceleration or deceleration, making them suitable for modeling scenarios like changes in speed or growth rates.
For instance, to predict future winning times for the 100-meter freestyle, as per our exercise, one would plug in the predicted year values into the corresponding \(t\) in our model equation \(y=86.24-0.752t+0.0037t^{2}\). To predict the time for 2008 (\(t=108\)) and 2012 (\(t=112\)), we substitute these values and solve for \(y\). The resulting \(y\) values are our predicted times. These predictions, albeit based on historic data, assume that the trend will continue in the same manner.
However, it's crucial to recognize that such predictions have limitations, especially as they extend far into the future. As we approach the limits of human potential and other influencing factors, the model's accuracy may decline. Hence, predictive modeling with quadratic equations is an iterative process, often requiring updates as new data becomes available.
For instance, to predict future winning times for the 100-meter freestyle, as per our exercise, one would plug in the predicted year values into the corresponding \(t\) in our model equation \(y=86.24-0.752t+0.0037t^{2}\). To predict the time for 2008 (\(t=108\)) and 2012 (\(t=112\)), we substitute these values and solve for \(y\). The resulting \(y\) values are our predicted times. These predictions, albeit based on historic data, assume that the trend will continue in the same manner.
However, it's crucial to recognize that such predictions have limitations, especially as they extend far into the future. As we approach the limits of human potential and other influencing factors, the model's accuracy may decline. Hence, predictive modeling with quadratic equations is an iterative process, often requiring updates as new data becomes available.
Horizontal Asymptote Analysis
Horizontal asymptotes are horizontal lines that a function approaches but never touches as the input grows increasingly large or small. They're commonly associated with rational functions rather than polynomials like the quadratic function we've used for the Olympic 100-meter freestyle prediction.
In our exercise, the quadratic model does not have a horizontal asymptote, as it's a characteristic not applicable to quadratic functions. Polynomials will either rise or fall without bound as \(x\) increases or decreases or will have a maximum or minimum point. This quadratic equation, then, will always produce a parabola that either opens upwards or downwards and does not approach a horizontal line.
However, for the practical application of modeling winning sprint times, it's logical to consider physical limitations, thus expecting the actual data to have a lower limit—or a 'practical' horizontal asymptote. This accounts for the inevitable plateau in athletic performance improvements due to human physiological limits. While the mathematical model does not inherently provide this, integrating such real-world considerations is key in developing more accurate and realistic models.
In our exercise, the quadratic model does not have a horizontal asymptote, as it's a characteristic not applicable to quadratic functions. Polynomials will either rise or fall without bound as \(x\) increases or decreases or will have a maximum or minimum point. This quadratic equation, then, will always produce a parabola that either opens upwards or downwards and does not approach a horizontal line.
However, for the practical application of modeling winning sprint times, it's logical to consider physical limitations, thus expecting the actual data to have a lower limit—or a 'practical' horizontal asymptote. This accounts for the inevitable plateau in athletic performance improvements due to human physiological limits. While the mathematical model does not inherently provide this, integrating such real-world considerations is key in developing more accurate and realistic models.
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