91Ó°ÊÓ

Heart health is a prime concern, because heart disease is the leading cause of death in the United States. Aerobic activities such as walking, jogging, and running are recommended for cardiovascular fitness, because they increase the heart's strength and stamina. a. A typical training recommendation for a beginner is to walk at a moderate pace of about 3.5 miles/hour (or approximately 0.0583 miles/minute) for 20 minutes. Construct a function that describes the distance traveled \(D_{\text {beginner }}\), in miles, as a function of time, \(T\), in minutes, for someone maintaining this pace. Construct a small table of values and graph the function using a reasonable domain. b. A more advanced training routine is to walk at a pace of 3.75 miles/hour (or 0.0625 miles/minute) for 10 minutes and then jog at 5.25 miles/hour (or 0.0875 miles/minute) for 10 minutes. Construct a piecewise linear function that gives the total distance, \(D_{\text {advanced }}\), as a function of time. \(T\), in minutes. Generate a small table of values and plot the graph of this function on your graph in part (a). c. Do these two graphs intersect? If so, what do the intersection point(s) represent?

Step by step solution

01

- Define the Distance Function for the Beginner

The distance traveled, denoted as \(D_{\text{beginner}}(T)\), can be defined as the product of the speed and the time. Given a speed of 0.0583 miles/minute and time \(T\) in minutes, the function is: \[D_{\text{beginner}}(T) = 0.0583T\]

02

- Create a Table of Values for the Beginner

Choose different values for time \(T\) such as 0, 5, 10, 15, and 20 minutes, and calculate the corresponding distance.\(T (minutes)\) | \(D_{\text{beginner}}(miles)\)0 | 05 | 0.291510 | 0.58315 | 0.874520 | 1.166

03

- Plot the Graph for the Beginner

Using the table values, plot the points on a graph with \(T\) on the horizontal axis and \(D_{\text{beginner}}\) on the vertical axis. Connect the points to form a straight line representing the function \(D_{\text{beginner}}(T) = 0.0583T\).

04

- Define the Piecewise Function for the Advanced Routine

For the advanced routine, we need to create a piecewise function based on different speeds for different time intervals. For \(0 \le T \le 10\), the speed is 0.0625 miles/minute, so \(D_{\text{advanced}}(T) = 0.0625T\). For \(10 < T \le 20\), the speed is 0.0875 miles/minute. The function is: \[D_{\text{advanced}}(T) = \begin{cases} 0.0625T & 0 \le T \le 10 \ 0.625 + 0.0875(T - 10) & 10 < T \le 20 \end{cases}\]

05

- Create a Table of Values for the Advanced Routine

Choose values for time \(T\) such as 0, 5, 10, 15, and 20 minutes, and calculate the corresponding distance.\(T (minutes)\) | \(D_{\text{advanced}}(miles)\)0 | 05 | 0.312510 | 0.62515 | 1.062520 | 1.5

06

- Plot the Graph for the Advanced Routine

Using the table values, plot the points on the same graph as in part (a) with \(T\) on the horizontal axis and \(D_{\text{advanced}}\) on the vertical axis. Connect the points to form two line segments representing the piecewise function \(D_{\text{advanced}}(T)\).

07

- Determine if the Graphs Intersect

Compare the graphs of \(D_{\text{beginner}}(T)\) and \(D_{\text{advanced}}(T)\). The graphs intersect if their values are equal for some \(T\) value. Set the equations equal to each other and solve for \(T\):\[0.0583T = 0.0625T \text{ (0 ≤ T ≤ 10)}\]. This has no solution within the given interval.\[0.0583T = 0.625 + 0.0875(T - 10) \text{ (10 < T ≤ 20)}\]. Solving this gives \(T ≈ 13.16\) minutes.

08

- Interpretation of Intersection Point

If the graphs intersect, the point of intersection represents the time at which both the beginner and advanced routines cover the same distance. For this exercise, the intersection occurs approximately at (13.16, 0.7672), meaning both routines cover about 0.7672 miles at around 13.16 minutes.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Cardiovascular Fitness

Cardiovascular fitness refers to the ability of the heart and lungs to supply oxygen-rich blood to the working muscles during physical activity. Activities like walking, jogging, and running are excellent for improving cardiovascular fitness. They help increase heart strength, endurance, and overall stamina. Consistent aerobic exercise can reduce the risk of heart disease, a leading cause of death in the US. For beginners, a moderate pace of walking at 3.5 miles per hour is recommended. As fitness improves, individuals can incorporate more intense activities, such as jogging or running.

Linear Functions

In mathematics, a linear function is a function that creates a straight line graph. It has the general form: \(y = mx + b\) where \(m\) represents the slope and \(b\) the y-intercept. For our exercise, the distance functions for both beginners and advanced routines are linear but serve different training intensities.
For beginners walking at 0.0583 miles per minute, the function is: \[D_{\text{beginner}}(T) = 0.0583T\] Here, the slope \(m = 0.0583\) reflects the speed, and \(T\) represents time in minutes.
For the advanced routine, the piecewise function reflects two linear segments:

Part 1: Walking for the first 10 minutes at a speed of 0.0625 miles/minute: \[D_{\text{advanced}}(T) = 0.0625T \quad \text{for} \quad 0 \le T \le 10\]

Part 2: Jogging for the next 10 minutes at a speed of 0.0875 miles/minute: \[D_{\text{advanced}}(T) = 0.625 + 0.0875(T - 10) \quad \text{for} \quad 10 < T \le 20\]

This piecewise function demonstrates how different activities can be combined to form a more complex training routine.

Function Graphs

Function graphs visually represent mathematical functions, showing how one quantity varies with another. Plotting functions for cardio routines helps compare distances covered over time for different training intensities.

For the beginner's routine, we plot: \[ \ D_{\text{beginner}}(T) \ = 0.0583T \,\] using values such as \(T = 0, 5, 10, 15, 20\) minutes. This results in a straight line.

For the advanced routine, we plot the piecewise function consisting of two line segments: \[ D_{\text{advanced}}(T) \ = \begin{cases} 0.0625T & 0 \le T \le 10\ \ \ 0.625 + 0.0875(T - 10) & 10 < T \le 20 \end{cases} \]

By comparing these graphs, we see that the advanced routine covers more distance over the same time intervals. These function graphs also help identify intersections, revealing points where both routines cover the same distance. The intersection at approximately \(T = 13.16\) minutes and distance \(D = 0.7672\) miles illustrates this.
Understanding graphs of functions can help students visualize and interpret distances covered in different cardiovascular training routines.