STEM 698 Mathematical Modeling
Homework Due Thursday, Oct. 18. Please type your responses on a separate sheet.
Read the excerpt from Thomas Hickey History of Twentieth-Century Philosophy of Science,
focusing on his summary of Kuhn’s book. Also read the very brief excerpt from Thomas Kuhn’s
Structure of Scientific Revolutions.
1. What is phlogiston? Why is it important in the context of Kuhn’s concept of scientific
revolutions?
2. In class, we discussed the discovery of dark matter, which is currently one of the most
important research pursuits in physics. In what way does the story of the discovery of dark
matter fit Kuhn’s hypothesis for how science works? Is there any way in which it doesn’t?
3. Discuss two ways in which Kuhn’s view of the nature of science differs fundamentally from
the points of view of some of other mathematicians and scientists whose writings we have
considered.
1. China, long the world’s most populous country, has a population of 1.312 billion. China has
worked hard to slow its growth rate. In 2007, China's population was growing at only 0.6%
annually. India, the world's second most populous country, has a population of 1.129
billion. India's population is growing at 1.6% annually.
Assuming that these growth rates continue, use exponential models to predict when India’s
population will exceed China’s.
2. For certain long periods of time, the US population is extremely well modeled by exponential
functions. One example is the period from 1800 to 1860.
Year
1800
1810
1820
1830
1840
1850
1860
Population
(in millions)
5.31
7.24
9.64
12.87
17.07
23.19
31.44
a. Find the best fit exponential model for this data. As we discussed in class, it is often
convenient to make the model with a variable where t = 0 in 1800.
b. Use the equation of the model to predict when the population of the US exceed 20 million.
c. Use the equation of the model to predict the population of the US in 1870.
d. The population of the US in 1870 was in fact 39.82 million. What might account for the poor
performance of the model after between 1860 and 1870?
3. (Forensics) Suppose at 6:00 PM a body is discovered in a basement of a building where the
ambient air temperature is maintained at a constant 72 degrees. At the moment of death, the body
temperature was 98.6 degrees, but after death the body cools, and eventually its temperature
matches the ambient air temperature. Beginning at 6:00 PM, the body temperature is measured:
Time since 6 PM
0
2
4
6
8
Temperature
(Degrees
Fahrenheit)
84.02
80.08
77.44
75.65
74.45
Using an exponential model, determine the approximate time of death. (Remember that
Newton’s Law of Cooling states the rate of change in temperature of an object is proportional to
the difference in temperature of the object and the surroundings. So to model this, you need to
calculate the difference between the temperature of the body and the surroundings.)
4. (Drug Metabolism) The drug valium is eliminated from the bloodstream at a rate of 1.9% per
hour.
a. Suppose a patient receives an initial dose of 20 milligrams. How much valium is in the
patient's blood 12 hours later?
b. What is the half-life of valium in the human body? (The half-life is the time it takes for the
amount to reach 50% of the original amount.)