Question

When a swimmer does the breaststroke, the function $$ \begin{aligned} S=&-241 t^7+1060 t^6-1870 t^5+1650 t^4 \\ &-737 t^3+144 t^2-2.43 t \end{aligned} $$ models the speed \(S\) (in meters per second) of the swimmer during one complete stroke, where \(t\) is the number of seconds since the start of the stroke and \(0 \leq t \leq 1.22\). Use a graphing calculator to graph the function. At what time during the stroke is the swimmer traveling the fastest?

Short answer

The exact time during the stroke when the swimmer is traveling at their fastest will depend on the specific graph of the function. After graphing the function, one must identify the maximum point over the interval \(0 \leq t \leq 1.22\) and report the corresponding \(t\) value.

Step-by-step solution

- Identify the problem

The function given is a polynomial expression representing the speed \(S\) of a swimmer at time \(t\) during a breaststroke. The task is to find when the swimmer is traveling the fastest during a stroke. This effectively means finding the maximum value of the function \(S(t)\) over the interval \(0 \leq t \leq 1.22\).

- Graph the function

Using a graphing calculator, software or online graphing tool, graph the function \(S(t) = −241 t^7 + 1060 t^6 − 1870 t^5 + 1650 t^4 − 737 t^3 + 144 t^2 − 2.43 t\). Make sure the graph covers the relevant time interval \(0 \leq t \leq 1.22\).

- Identify the maximum value

After graphing the function, identify the maximum value on the graph in the range \(0 \leq t \leq 1.22\). The \(t\) value corresponding to this maximum value is the time when the swimmer is moving at their fastest speed during the stroke.

Key concepts

Graphing Calculator

In math and science, a graphing calculator is an essential tool that helps visualize complex functions, like polynomial equations. These calculators, either physical or online, allow you to plot functions on a graph to see their behavior over a certain range.
To graph the function given in the exercise, you can use a graphing calculator by entering the polynomial expression directly.

• Ensure your calculator covers the correct time frame of the polynomial, in this case, from 0 to 1.22 seconds.
• This will help you to see the function's progression over time accurately.

Using the graphing tool: - Enter the polynomial function exactly as it appears. - Look for a settings option on your calculator; set the x-axis to represent time from 0 to 1.22. - Most calculators will allow you to easily adjust this range. - Set the y-axis to cover expected speed values. This varies, so you might need to zoom in or out to see the function clearly. Graphing calculators are particularly useful with complex functions because they provide visual insights. Once plotted, it becomes easier to spot maximums, minimums, and other critical points.

Maximum Value

In mathematics, finding the maximum value of a function is crucial, particularly when dealing with real-world scenarios like the speed of a swimmer. The maximum point of a function is where it reaches its highest value within the specified range. In this case, the polynomial function models the swimmer's speed.
To identify the maximum value on a graph, follow these steps:

• Once you graph the function, observe the curve within the interval from the start to the end of the stroke, 0 to 1.22 seconds.
• Look for the peak or the highest point on the curve, as this indicates the maximum speed.
• Many graphing calculators have a built-in feature to find the maximum or minimum values. This can be used to determine the exact point of maximum speed.

Knowing the maximum value—both when it occurs and its magnitude—gives insights into performance during various activities.

Speed and Time Analysis

Analyzing speed over time is an essential aspect of understanding motion dynamics. The function presented models the speed of a swimmer doing the breaststroke over time. In analyzing such a function, you explore how the swimmer's speed changes at different points during the stroke. This involves both qualitative and quantitative analysis.
Qualitatively, the graph of the function shows the evolution of speed:

• Initially, the speed might rise as the swimmer accelerates.
• Next, it reaches a peak—the fastest point—before slowing down.

Quantitatively, you can calculate exactly when and how fast the swimmer is moving by analyzing the function. Each part of the polynomial influences the shape and behavior of the graph. Evaluating these shifts lets you measure performance aspects, such as timing and pacing. By understanding the peak times of speed, strategies can be devised to optimize performance during a race or workout.