Getting Fit with Mathematics

 

(This project is adapted from the Interdisciplinary Lively Application Project written by Joseph Myers, Walter Barge, Todd Crowder, and Kathleen Snook at West Point.)

 

Background

 

The Physical Education Department has been doing some fitness studies, and has asked you to help them interpret their data.  So that you will understand the context of the data, they give you the following background information.

 

The ability to sustain a high level of physical activity without undue fatigue depends on two factors:  (1) oxygen delivery and (2) the capacity of specific muscle cells to generate the cellular fuel adenosine triphosphate, or ATP [M, pg. 223].  The formation of ATP to be used for muscular energy begins when glucose molecules undergo a chemical transformation in a process known as glycolysis.  When the body is subject to light exercise, even of a long duration, the ATP is produced through an efficient slow glycolysis.  Waste products from the glycolysis are easily removed from muscles at about the same rate that they are produced, and there is little accumulation.  Under conditions of strenuous exercise, the demand for ATP can exceed the cell's ability to produce it efficiently.  When this happens, the muscle cells resort to an inefficient fast glycolysis which releases waste products faster than thay can be removed from the muscles.  The result is an accumulation of waste products in the muscle fibers and blood stream.  One of these waste products is lactic acid.  The fast glycolysis buys time for the muscles by rapidly producing ATP even if the oxygen supply is inadequate or the exercise too strenuous to produce ATP efficiently [M, pg. 125].  However, using fast glycolysis to meet muscular energy demands is only a temporary solution; the accumulation of lactic acid in the blood stream contributes to muscle fatigue which prevents continued physical activity.  Effective aerobic conditioning is one way to enhance the capacity of specific muscle cells to generate ATP efficiently and thus delay muscle fatigue.

 

The capacity of oxygen (O2) consumption is a fundamental measure of maximal aerobic power [M, pg. 211].  The highest rate of oxygen consumption during a controlled fitness test is called max VO2 (i.e., Value of O2), and is measured in liters/minute of oxygen consumed.  No one can operate for a long time one's max VO2 level, but there is a significant connection between a person's max VO2 and the formation of lactic acid in the blood.  For people of all levels of physical fitness, when engaging in strenuous physical activity there is a certain percentage of max VO2 at which the production of blood lactic acid shows a near exponential increase.  This point is called the Onset of Blood Lactic Acid (OBLA) [M, pg. 126].  As stated above, endurance is influenced by the oxygen delivery rate, which is linked to max VO2, and by the generation of ATP, which is linked to the point where OBLA begins.

 

[M]  McArdle, W.D., Katch, F. and Katch, V.L., Exercise Physiology.  Malvern, PA.  Lea and Febiger.  1991.

 

Data

 

The Physical Education Department has provided you with the following tables recording relationships between exercise duration, power exerted, oxygen consumption rate, and lactic acid release rate.  The tables were produced by averaging data collected during student fitness tests.

 

Table 1:  Fitness tests were done on a treadmill.  Initially, the treadmill is horizontal and moving at an easy pace.  Over time, the treadmill is gradually inclined and the treadmill belt speed is gradually increased (both at a constant rate of increase).  The exerciser must use more and more power to "keep up with" and "stay on" the treadmill.

 

Time (minutes)

Power (watts)

0

0

1

14

2

28

3

42

4

56

5

70

6

84

7

98

8

112

9

126

10

140

11

154

12

168

13

182

14

196

15

210

16

224

 

Table 2:  The rate at which oxygen is consumed during exercise is a function of the power being expended at that instant.  When power is low, oxygen consumption is low.  As power increases (as the treadmill test becomes more difficult), the oxygen consumption rate increases up to the max VO2.  The test is concluded when the runner reaches max VO2 and can no longer stay on the treadmill.  The average max VO2 in the student fitness tests was 5.05 liters/minute.

 

Power (watts)

Oxygen Consumption Rate

(liters/minute)

0

0

20

0.90

40

1.75

60

2.50

80

3.20

100

3.70

120

4.20

140

4.45

160

4.60

180

4.75

200

4.95

220

5.05

 

Table 3:  Lactic acid formation in the blood is a function of the rate of oxygen consumption.  Some lactic acid is produced even when the oxygen consumption rate is low.  Notice that when the onset of blood lactic acid (OBLA) begins (when the oxygen consumption rate rises above 3 liters/minute), the concentration of lactic acid in the blood increases dramatically.

 

Oxygen Consumption Rate

(liters/minute)

Lactic Acid Released

(millimoles/minute)

0

1.000

0.5

1.125

1

1.250

1.5

1.375

2

1.500

2.5

2.250

3

3.000

3.5

7.000

4

11.000

 

Problems

 

Remember that your results will be submitted as a report to the Physical Education Department.  Where appropriate, please write your answers using complete sentences, correct grammar, and good style.

 

1.      Plot the data in the three tables.  For each plot, say whether it is linear or non-linear (and explain your choice), where it is increasing or decreasing, and how the rate of increase changes over time.

2.      Find a line which models the data in Table 1 as accurately as possible.  Describe how well your model fits the data (qualitatively).

3.      Find a quadratic function which models the data in Table 2.  Describe how well your model fits the data (qualitatively).

4.      Using Tables 1 and 2, produce a table for oxygen consumption as a function of time.  Plot this new table, and describe it as in problem 1.

5.      Find a quadratic function which models the data from problem 4. Describe how well your model fits the data (qualitatively).

6.      Using your functions from problems 2 and 3, find another quadratic function which models the data.  Compare this to your answer to problem 5.  Which model fits better?

7.      How long did the average student in the study stay on the treadmill?

8.      Model the data in Table 3 by a piecewise defined function consisting of three linear parts.

9.      Using your answer in problem 4 and Table 3, produce a table for lactic acid released as a function of time.  Plot this table, and describe it as in problem 1.

10.  When did the average student in the study reach OBLA?  What power were they exerting at this time?  What was their rate of oxygen consumption?

 

Graded out of 35 points