Problem 89 - Calculus of a Single Variable

Question

Modeling Data The average typing speeds \(S\) (in words per minute) of a typing student after \(t\) weeks of lessons are shown in the table. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline t & {5} & {10} & {15} & {20} & {25} & {30} \\ \hline S & {28} & {56} & {79} & {90} & {93} & {94} \\\ \hline\end{array} $$ A model for the data is \(S=\frac{100 t^{2}}{65+t^{2}}, t>0\) (a) Use a graphing utility to plot the data and graph the model. (b) Does there appear to be a limiting typing speed? Explain.

Step-by-Step Solution

Answer

The graphing of the given data and model will provide a visual representation of the relationship between typing speed and weeks of lessons. If the graph of the model appears to approach a certain speed but never exceed it (represented by a horizontal asymptote), it can be said that there is a limiting typing speed. The limit value would be represented by the y-coordinate of the horizontal asymptote.

Step 1: Plot the given data

Plot the given (\(t, S\)) points on a graph. Use weeks (t) for the x-axis and typing speed (S) for the y-axis.

Step 2: Graph the mathematical model

Next, plot the given mathematical model \(S=\frac{100 t^{2}}{65+t^{2}}\) on the same graph, using a different color line for differentiation. This model should overlay the dataset points if it correctly represents them.

Step 3: Analyzing the Graph

Inspect the graph visually to investigate if there's a limiting typing speed. This would likely appear as a horizontal asymptote, where the modelled function appears to approach a certain speed (S-value) but never exceeds it.

Step 4: Interpret the Graph

The vertical asymptote represents the limit of typing speed that the model predicts. A horizontal asymptote indicates a limit to the average typing speed regardless of the number of lessons. If such an asymptote is observed in the graph, this suggests a limiting typing speed indeed exists. The approximate value of this limit is the y-coordinate of the asymptote.

Key Concepts

Average Typing Speed

Average typing speed refers to the number of words a person can type in a minute after practicing over a specific period. For the data in the table, this is calculated by recording typing speeds at different intervals of weeks, showing how a student's skill progresses with time. As one practices more, generally, the typing speed should improve.

But it's interesting to note that there may be a point when improvements slow down, suggesting a maximum achievable speed. This can be seen when examining trends in the data over time, such as the initial sharp increase in typing speed that tapers off, indicating reaching a natural limit.

Graphing Utility

A graphing utility is a tool used to create visual representations of mathematical data and models. These can be physical tools like graph paper and rulers or digital ones such as software and online graphing tools. They allow us to input data points and equations to create a visual that helps to better understand relationships and trends in the data.

For this exercise, the graphing utility helps us plot the given time and speed data points, and also the mathematical model function. The visualization aids in comparing the actual data against the model, revealing how well the model represents the actual typing speeds over time.

Mathematical Model

A mathematical model is an equation or formula that describes real-world phenomena using mathematics. In this exercise, the given model is expressed as \(S=\frac{100 t^{2}}{65+t^{2}}\), aiming to describe the relationship between weeks of practice \(t\) and the average typing speed \(S\).

The purpose of using a model is to predict future outcomes or understand past data trends by associating variables with measurable quantities. Here, the model helps predict average typing speeds given any number of weeks of lessons, allowing us to visualize trends and predict when someone's typing improvements might level off.

Horizontal Asymptote

A horizontal asymptote in graphing is a horizontal line that a curve approaches as it heads towards infinity. It suggests a limit to how far a value, such as typing speed, can increase regardless of time spent practicing.

In the context of the typing speed model, the horizontal asymptote reflects the maximum speed a student can achieve with unlimited practice. According to the graph derived from the model \(S=\frac{100 t^{2}}{65+t^{2}}\), the typing speed approaches a limit but never quite reaches or exceeds it. Observing where the curve flattens gives us this asymptote, indicating an ultimate speed ceiling for the typing student.