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The function P(t) = 145e-0.092t models a runner’s pulse, P(t), in beats per minute, t minutes after a race, where 0 ≤ t ≤15. Graph the function using a graphing utility. TRACE along the graph and 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.

1)Use this model and the formula for Sn to find the total cost of tuition and fees at public colleges for a four-year period from the school year ending in 2007 through the school year ending in 2010.

2)How does this compare with the actual sum you obtained in part (a)?

## Modeling a Runner's Pulse

The downward sloping of the graph from the right to the left shows that there is a negative relationship between the pulse beat of the runner (heart rate), P(t) and the time taken, t in minutes, such that as the time, t in minutes increases ,there is a reduction in the rate of runner’s pulse beat ( heart rate) ,P(t) as suggested by the model function (Few, 2008).

At a point where the pulse beat (heart rate) of a runner is at a rate 70 beats for every minute, the time, t in minutes will be 7.9137 minutes which can be approximated to be 8 minutes. This can also be confirmed using the function, as;

Which can be rounded off to 70 pulse beats (heart rate) after 8 minutes. But if time, t in minutes is taken to be equal to 8 minutes then the results of the number of pulse beat will be slightly different as shown below.

Which can still also be rounded off to 70 pulse beat after 8 minutes.

A runner who participates in long distance races such as marathon has a low resting heart rate which means that a person who is just resting or seated has a slower pulse beat (heart rate) as compared to the heart of a normal of a normal person (Kholsa & Campbell , 2002). A normal person has a resting pulse which is between 70 pulse beat per minute (heart rate) to 75 pulse beat per minute (heart rate) which is what happens at time, t = 8 minutes for the case of the model function which suggest that the runner is resting.

An athlete who trains a lot can be able to reduce the pulse beat per minute (heart rate) of a normal person to its half. This is the case for this graph at time, t = 15, where pulse beat per minute, P(t) is equals to 36 pulse beat per minute which is now half the normal pulse beat for a normal person (Laird & Campbell, 2008). This shows that the runner in the graph function for the model is an athlete who trains a lot and therefore as the time, t in minutes increases (as the runner continues with training), the pulse beat per minute (the rate of heart beat) decreases to half the pulse beat per minute (heart rate) at the start time, t equals to zero which shows that as exercise training increases the volume of oxygen that is delivered to the muscles for every pulse beat, hence the heart needs to beats less to supply the same volume of oxygen needed at time, t equals to zero when the runner was resting.

## Calculating Tuition and Fees at Public Colleges

The average pulse beat of an athlete in during a marathon is around 160 pulse beat per minute for a person who is aged 20’s and has a resting heart rate of 55 pulse beat per minute and a maximum heart rate of 200 pulse beat per minute which is in most cases determined by the age of the person (Laird & Campbell, 2008). The maximum heart rate (pulse beat per minute) reduces as the person’s age increases (as the person grows older). This can also be the case for the graphical function since as the time, t tends towards t = 15, the pulse beat per minute, P(t) (heart rate) reduces drastically (tends towards zero). This shows that as the runner continues running, the runner’s pulse beat per minute (heart rate) reduces as a result of the increase in age of the runner.

**Solution**

For the period of four years, the sum of school fees paid to institutions owned by the government, which includes the money payable for tuition and other levies as required by the individual institutions ranging from the academic years (period when students are in sessions in those institutions) 2006/2007, 2007/2008, 2008/2009 and 2009/2010 as per the bar graph is given as follows (Shepherd, 2003):

Where;

is the sum of the school fees paid to government owned institutions (includes tuition fees, examination fees and other levies as required by the individual institutions) for all the academic years which ends in the years 2007,2008,2009 and 2010 (Brickley, Smith, & Zimmerman, 2001).

t is the period or time between the academic years which ends in the years 2007, 2008, 2009 and 2010 for the institutions owned by government.

T & F is the individual sum of the school fees paid to government owned institutions (includes tuition fees, examination fees and other levies as required by the individual institutions) for the academic years which ends in the years 2007, 2008, 2009 and 2010 for the institutions owned by government

The mathematical model function shows the school fees paid by the students to government owned institutions (includes tuition fees, examination fees and other levies as required by the individual institutions) for the academic year n, where n = 1,2,3 and 4 denotes the academic years for the institutions which ends in the years 2007, 2008, 2009 and 2010 respectively.

Using this mathematical model function, the school fees paid to government owned institutions (includes tuition fees, examination fees and other levies as required by the individual institutions) for the period of four years; (the academic years for the institutions which ends in the years 2007, 2008, 2009 and 2010), taking n to be equal to 1, 2, 3 and 4 respectively will be given as:

## Observations and Analysis

The sum of the school fees paid to government owned institutions (includes tuition fees, examination fees and other levies as required by the individual institutions) for all the academic years which ends in the years 2007,2008,2009 and 2010 for the values found using the mathematical model function is less than the actual sum of school fees paid to institutions owned by the government, which includes the money payable for tuition and other levies as required by the individual institutions ranging from the academic years (period when students are in sessions in those institutions) for the period of four years as per the bar graph by 10.

The difference of 10 between the sum of the school fees paid to government owned institutions gotten from the bar graph and sum of the school fees paid to government owned institutions (includes tuition fees, examination fees and other levies as required by the individual institutions) gotten from the model for the public colleges in the four academic years is due to the error incurred when using the mathematical model function to calculate the sum of the school fees paid to government owned institutions for all the academic years which ends in the years 2007,2008,2009 and 2010.

The individual errors incurred in each of the period of time between the academic years which ends in the years 2007, 2008, 2009 and 2010 for the institutions owned by government, are then transferred to the sum of the fees while calculating the sum of the fees paid to government owned institutions for the period of four years required (Shepherd, 2003).

If the sum of the school fees paid to the government institutions for the individual years from the mathematical model function are compared to the corresponding sum of the school fees paid to government owned institutions (schools) from the bar graph on the same year, the errors for the individual years alternates from positive to negative (Stock & Watson, 2007). For example, the difference between the costs of tuition and fees from the bar graph and the model for the year ending in 2007 is positive in 22, for the year ending in 2008 is negative 24, for the year ending in 2009 is negative 19 and for the year ending in 2010 is positive 21. This shows that the error is random and therefore it cannot be predicted because it changes from one period to another. The difference between the sum of school fees paid to government owned institutions gotten from the bar graph and sum of school fees paid to government owned institutions calculated from the mathematical model function which accounts for the total error is therefore the sum of individuals errors from the individual years for the period of four years (2007, 2008, 2009 and 2010). The error cannot therefore be taken care of by a constant number since we cannot predict the exact value of the error for the consecutive years since they change randomly and whether the error will be negative or positive since that will be determined by the mathematical model .

References

- Brickley, J. A., Smith, C. W., & Zimmerman, J. L. (2001). Managerial Economics and Organizational architecture. New York: McGraw-Hill Irwin.
- Few, S. (2008). Perceptual Edge. Line Graphs and Irregular Intervals: An Incomplete Partnership, 3-10.
- Kholsa, T., & Campbell , H. (2002). Resting Pulse rate in marathon runners. PMC Article, 1444.
- Laird, G. D., & Campbell, M. J. (2008). Exercise Levels and Resting Pulse rate in the Community. British Journal of Sports Medicine, 148-152.
- Meta Calculator. (2018, 11 24). Meta Calculator. Retrieved from Meta Calculator: https://www.meta-calculator.com/?panel-101-equations&data-bounds-xMin=-15&data-bounds-xMax=15&data-bounds-yMin=-11&data-bounds-yMax=11&data-equations-0=%22y%3D145*exp(-0.092%20*%20x)%22&data-domains-0=%5B0%2C15%5D&data-ranges-0=%5B0%2C145%5D&data-rand=und
- Nation, R. D., & Aufmann , R. N. (2007). College Algebra.London: Chegg Study.
- Shepherd, W. R. (2003). Cost and Production Functions.Princeton: Princeton University Press.
- Stock, J. H., & Watson, M. W. (2007). Introduction to Econometrics.Boston: MA: Addison-Wesley.

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