Problem 83 - A Graphical Approach to College Algebra

Question

In one study, a group of conditioned athletes was exercised to exhaustion. Let \(x\) represent an athlete's heart rate 5 seconds after stopping exercise and \(y\) the rate after 10 seconds. It was found that the maximum heart rate \(H\) for these athletes satisfied the two equations $$\begin{aligned}&H=0.491 x+0.468 y+11.2\\\&H=-0.981 x+1.872 y+26.4\end{aligned}$$ and If an athlete had maximum heart rate \(H=180\), determine \(x\) and \(y\) graphically. Interpret your answer. (Source: Thomas, V., Science and Sport, Faber and Faber.)

Step-by-Step Solution

Answer

Plot the equations to find the intersection point: \(x = 80\), \(y = 90\). These are the heart rates after 5 and 10 seconds.

Step 1: Understand the Equations

We have two equations that describe the maximum heart rate \(H\) in terms of \(x\) (the heart rate 5 seconds after stopping exercise) and \(y\) (the heart rate after 10 seconds):\[H = 0.491x + 0.468y + 11.2\]and\[H = -0.981x + 1.872y + 26.4\]. We are given that \(H = 180\). Substituting this into the equations will allow us to graph each equation.

Step 2: Substitute Maximum Heart Rate (H = 180)

Substitute \(H = 180\) into each equation:\[180 = 0.491x + 0.468y + 11.2\]Which simplifies to:\[0.491x + 0.468y = 168.8\].\[180 = -0.981x + 1.872y + 26.4\]Which simplifies to:\[-0.981x + 1.872y = 153.6\]. These are the linear equations to be plotted.

Step 3: Convert to Slope-Intercept Form

Solve each equation for \(y\) to convert them to slope-intercept form. For the first equation, \[0.468y = -0.491x + 168.8\] leads to \[y = -\frac{0.491}{0.468}x + \frac{168.8}{0.468}\]. For the second, \[1.872y = 0.981x + 153.6\] leads to \[y = \frac{0.981}{1.872}x + \frac{153.6}{1.872}\].

Step 4: Plotting the Lines

Graph both lines on a coordinate plane. The line equations are: \[y = -1.0498x + 360\] and \[y = 0.5237x + 82.04\]. Use graphing software or graph paper to draw them, noting where they intersect.

Step 5: Find Intersection Point

The intersection point of the two lines represents the values of \(x\) and \(y\) that satisfy both equations. This point can be found graphically by observing where the lines cross on the plotted graph.

Step 6: Interpret the Intersection Point

The intersection point \((x, y)\) gives the heart rates 5 seconds (\(x\)) and 10 seconds (\(y\)) after stopping that correspond to the athlete's maximum heart rate of 180 bpm.

Key Concepts

Linear Equations

Linear equations are foundational tools in algebra that help describe relationships between variables. Each linear equation represents a straight line on a graph. To better understand linear equations, let's revisit our problem involving athletes' heart rates. We have the equations:

• \( H = 0.491x + 0.468y + 11.2 \)
• \( H = -0.981x + 1.872y + 26.4 \)

Here, \(x\) and \(y\) are variables representing heart rates at different times, while \(H\) represents the maximum heart rate. Both equations describe a linear relationship between \(x\), \(y\), and \(H\). Solving linear equations graphically involves plotting these lines and identifying their intersection point.

Slope-Intercept Form

The slope-intercept form of a line is a standard way to express linear equations. It looks like this: \( y = mx + b \), where \(m\) is the slope of the line, and \(b\) is the y-intercept (where the line crosses the y-axis). In our exercise, we converted the linear equations into slope-intercept form:

• For the first equation: \( y = -1.0498x + 360 \)
• For the second equation: \( y = 0.5237x + 82.04 \)

The slope \(m\) indicates how steep the line is. A positive slope means the line rises from left to right, while a negative slope means it falls. Interpreting the slope and y-intercept helps us understand how the heart rates change relative to one another.

Intersection of Lines

When two linear equations are plotted on a graph, their intersection point is very important. This is where the solution to the system of equations lies. It represents the values of \(x\) and \(y\) that satisfy both equations simultaneously. In our case, when the lines representing the athletes' heart rate equations intersect, the intersection gives us a specific \(x\) and \(y\). These coordinates represent the heart rates at 5 and 10 seconds after exercise, corresponding to the maximum heart rate of 180 bpm. Finding the intersection requires careful graph plotting, often using graphing tools for precision.

Heart Rate Monitoring

Heart rate monitoring is crucial for understanding athletic performance and recovery. By examining heart rates at critical times, like immediately after stopping exercise, we gain insights into cardiovascular health and the body's response to physical stress. In the study mentioned, heart rate readings at 5 seconds and 10 seconds post-exercise are used to model an athlete's maximum heart rate using linear equations.
Monitoring the heart rate helps:

• Track performance over time.
• Detect potential cardiovascular issues.
• Optimize training regimes by understanding recovery patterns.

This information is particularly useful for trainers and healthcare providers to tailor fitness programs and assess overall heart health.