Assignment 04 Ma240 College Algebra Directions Be Sur 069093
Assignment 04ma240 College Algebradirections Be Sure To Save An Elect
Assignment 04ma240 College Algebra directions: Be sure to save an electronic copy of your answer before submitting it for grading. Answer in complete sentences, using correct English, spelling, and grammar. Sources must be cited in APA format. Your response should be four (4) double-spaced pages; refer to the "Assignment Format" page on the Course Home page for specific formatting requirements. The function P(t) = 145 e^{-0.092 t} models a runner’s pulse in beats per minute after t minutes into a race, where 0 ≤ t ≤ 15.
Graph the function using a graphing utility. Trace along the graph to determine after how many minutes the runner’s pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically. This is the end of Assignment 4.
Paper For Above instruction
Understanding the dynamics of a runner’s pulse during a race is essential for monitoring cardiovascular stress and ensuring optimal performance. The provided function, P(t) = 145 e^{-0.092 t}, models the runner’s pulse rate over time, with t representing minutes after the start of the race. Analyzing this exponential decay function allows us to interpret how the pulse decreases from an initial rate and at what point it reaches a specific threshold, in this case, 70 beats per minute.
Graphing the Function
To comprehend the behavior of P(t), the first step involves graphing the function using a graphing calculator or software such as Desmos, GeoGebra, or a graphing calculator. The graph of P(t) will illustrate an exponential decay from an initial value of 145 beats per minute at t = 0, decreasing as time progresses due to the physical cooling and recovery process post-exercise.
On graphing, we observe that at t=0, the pulse rate is at its maximum, 145 bpm, gradually decreasing as t increases, asymptotically approaching near the baseline levels. Tracing the graph along its curve allows us to locate the point where P(t) intersects the line y=70, indicating the time when the pulse drops to 70 bpm.
Determining the Time Algebraically
To corroborate the graphical observation, we solve for t algebraically when P(t) = 70:
Set P(t) = 70:
145 e^{-0.092 t} = 70
Divide both sides by 145:
e^{-0.092 t} = \frac{70}{145} = \frac{14}{29} \approx 0.4828
Apply the natural logarithm to both sides:
\ln e^{-0.092 t} = \ln 0.4828
-0.092 t = \ln 0.4828
Calculate \ln 0.4828:
\ln 0.4828 \approx -0.7289
Solve for t:
t = \frac{-0.7289}{-0.092} \approx 7.921
Thus, the runner’s pulse reaches 70 bpm approximately at t ≈ 7.9 minutes.
Discussion
The exponential decay model captures the physiological process of pulse recovery after exertion. Notably, the time of approximately 7.9 minutes aligns with typical recovery patterns observed in runners, where pulse rates gradually return to resting or near-resting levels within ten minutes after sprinting or intense activity. The graphical and algebraic analysis reinforce each other, demonstrating the reliability of the model and its utility in athletic training and health monitoring contexts.
Implications and Applications
Understanding this decay can assist coaches and athletes in designing training programs that optimize recovery times and prevent overexertion. Moreover, such models may be used to compare individual recovery patterns or to evaluate the effectiveness of different training regimens. In health contexts, tracking pulse recovery times can alert medical professionals to abnormal cardiovascular responses, potentially indicating underlying health issues.
Conclusion
Through creating a graph and solving algebraically, we find that a runner’s pulse rate decreases from 145 bpm to 70 bpm approximately after 7.9 minutes. This combined approach demonstrates how mathematical modeling, visualization, and algebraic solutions converge to provide meaningful insights into physiological processes. Such analytical techniques are invaluable in sports science, health monitoring, and related fields, illustrating the practical applications of exponential functions in real-world scenarios.
References
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