Greenwood press

EXPONENTIAL GROWTH
31
The growth of the rabbit population in
Australia from 1859 –1864 modeled
by an exponential growth equation.
A graph depicting growth of the
rabbit population in Australia
from 1859 –1864.
six-month
year
periods (x)
rabbit population (y)
1859
0
24
1
24 • 2 = 48
1860
2
24 • 2 • 2 = 96
3
24 • 2 • 2 • 2 = 192
1861
4
24 • 2
4
= 384
5
24 • 2
5
= 768
1862
6
24 • 2
6
= 1,536
7
24 • 2
7
= 3,072
1863
8
24 • 2
8
= 6,144
9
24 • 2
9
= 12,288
1864
10
24 • 2
10
= 24,576

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pal held for the customer. The return addition of interest payments to the princi-
pal so that the interest amount can earn interest in later years is called compound
interest. The growth factor in compound-interest problems is 1 plus the annual
yield. So an investor who buys a $5,000 CD advertised at 6.5 percent annual
yield will receive 
5000(1 + .065)
x
after
x years. After three years, this CD
would be valued at 
5000(1.065)
3
= $6,039.75. Banks may choose to compound
interest more frequently. The banking version of the exponential growth formula
is
A = P (1 + r/n)
nt
, where 
A is the amount at the end of t years, P is the start-
ing principal, 
r is the stated interest rate, and n is the number of periods per year
that interest will be compounded. A typical CD will have interest compounded
each quarter. Financial institutions can offer more-frequent compounding, such
as monthly or daily. Some even offer continuous compounding, which has the
formula
A = P e
rt
, where 
A is the value of the investment at time t, P is the ini-
tial principal, 
r is the interest rate, and e ≈ 2.7183. For a given interest rate, more
frequent compounding yields a higher return, but that return does not increase
dramatically as the compounding period moves from months to days to continu-
ous. Because the number of compounding periods can affect the rate of return on
an investment, federal law requires financial institutions to state the annual yield
as well as an interest rate so that consumers can make easier comparisons among
investment opportunities.
Benjamin Franklin was one of the pioneers in the use of exponential growth
models for money and population. In 1790, Franklin established a trust of
$8,000. He specified that his investment should be compounded annually for 200
years, at which time the funds should be split evenly between the cities of
Philadelphia and Boston, and used for loans to “young apprentices like himself.”
Franklin anticipated that the fund would be worth $20.3 million after 200 years
if the annual yield averaged 4 percent. However, the annual yield averaged about
3.4 percent, so $6.5 million was in the fund when it was dispersed to the two
cities in 1990. 
Franklin established the practice of studying the American population by
using exponential growth. He recognized that the warning of the Englishman
Thomas Malthus (1766 –1834) that population under exponential growth would
outstrip food sources might apply to the new country of the United States. Frank-
lin urged that the growth of states and the entire country be tracked each year.
Some historians contend that President Lincoln used exponential growth models
70 years after Franklin’s recommendation. Lincoln used censuses from 1790 to
1860 to predict that the population of the United States would be over 250 mil-
lion in 1930. The population did not reach this figure until 1990. This shows that
exponential functions can describe situations only as long as the growth factor
remains constant. There are many factors such as economics, war, and disease
that can affect the rate of population growth. 
When the Center for Disease Control identifies a new epidemic of flu, expo-
nential growth functions describe the numbers of early cases of infection quite
well. A good definition of epidemic is a situation in which cases of disease in-
crease exponentially. However, as people build up immunization, the disease