The Number e and Natural Logarithms
Convert the following functions into the form P = P 0at.
1. P = 15e0.25t
2. P = 2e-0.5t
Convert the following functions into the form P = P 0ekt.
3. P = 10(1.7)t
4. P = 174(0.9)t
5. a) A population grows according to the equation P = P 0ekt. Find the population as
a function of time, t, if it grows at a continuous rate of 2% a year and starts at 1
b) Plot a graph of the population against time.
6. An almanac lists the population of the earth in 1980 as 4.478 billion and in 1994 as
5.642 billion. Assuming it increases exponentially, predict the earth’s population in
the year 2010.
7. One of the main contaminants of a nuclear accident, such as that at Chernobyl, is
strontium-90, which decays exponentially at a continuous rate of approximately
2.47% per year. Preliminary estimates after the Chernobyl disaster suggested that
it would be about 100 years before the region would again be safe for human
habitation. What percent of the original strontium-90 would still remain at this time?
8. The population, P, of Nicaragua was 3.6 million in 1990 and growing at an annual
rate of 3.4%. Let t be time in years since 1990.
a) Express P as a function in the form P = P 0at.
b) Express P as an exponential function using the base e.
c) Compare the annual and continuous growth rates.