POWERING & POPULATION GROWTH
An important application of powers is in population growth situations. As
an example, consider rabbit populations, which can grow quickly. In 1859
in Australia, 22 rabbits were imported from Europe as new source of food.
Rabbits are not native to Australia, but conditions there were ideal for
rabbits, and they flourished. Soon there were so many rabbits that they
damaged grazing land. By 1889, the government was offering a reward
for a way to control the rabbit population.
Example 1
Twenty-five rabbits are introduced to an area. Assume that the rabbit
population doubles every six months. How many rabbits will there be after
5 years?
Solution
Since the population doubles twice each year, in 5 years it will double 10
times. The number of rabbits will be
25∙2∙2∙2∙2∙2∙2∙2∙2∙2∙2
To evaluate this expression on a calculator rewrite it as
25 ∙210
Use the yx or ˆ key. After 5 years there will be 25,600 rabbits.
WHAT IS EXPONENTIAL GROWTH?
The rabbit population in example 1 is said to grow exponentially. In
exponential growth, the original amount is repeatedly multiplied by a
positive number called the growth factor.
GROWTH MODEL FOR POWERING
If a quantity is multiplied by a positive number g (the growth factor) in
each of x time periods, then after x periods, the quantity will be multiplied
by gx.
In example 1, the population doubles (is multiplied by 2) every six months,
so g=2. There are 10 time periods (think 5 years = 10 half years), so x = 10.
The original number of rabbits, 25, is multiplied by gx, or 210.
WHAT HAPPENS IF THE EXPONENT IS ZERO?
In the growth model, x can be any real number. Consider the statement
of the growth model when x = 0. It reads:
If the quantity is multiplied by a positive number g in each of 0 time
periods, then after the 0 time periods, the quantity will be multiplied by g0.
In 0 time periods, no time can elapse. Thus the quantity remains the
same. It can remain the same only if it is multiplied by 1. This means that
g0 = 1, regardless of the value of the growth factor g. this property applies
also when g is a negative number.
ZERO EXPONENT PROPERTY
If g is any nonzero real number, then g0 = 1.
In words, the zero power of any nonzero number equals 1. For example,
40 =1, (-2)0 =1, and (5/7)0 =1. The zero power of 0, which would be written
00, is undefined.
WHAT DOES A GRAPH OF EXPONENTIAL GROWTH LOOK LIKE?
An equation of the form y=b∙gx, where g is a number greater than 1, can
describe exponential growth. Graphs of such equations are not lines.
They are exponential growth curves.
Example 2
Graph the equation y=3.2x, when x is 0, 1, 2, 3, 4.
Solution
Substitute x=0, 1, 2, 3, 4 into the formula y=3.2x. Below we show the
computation and the results listed as (x, y) pairs.
3∙20 = 3∙1 = 3
3∙21 = 3∙2 = 6
3∙22 = 3∙4 = 12
3∙23 = 3∙8 = 24
3∙24 = 3∙16 = 48
X
0
1
2
3
4
y=3.2x
3
6
12
24
48
Graph of y=3.2x
60
50
y
40
30
Series1
20
10
0
1
2
3
4
5
x
AN EXAMPLE OF EXPONENTIAL POPULATION GROWTH
Other than the money calculated using compound interest, few things in
the real world grow exactly exponentially. Over the short term, however,
exponential growth approximates the changes that have been observed
in many different populations. For instance, consider the population of
California from 1930 to 1990.
Census
1930
1940
1950
1960
1970
1980
1990
Population
5,677,251
6,907,387
10,586,223
15,717,204
19,971,069
23,667,764
29,760,021
To see how close California’s population growth is to exponential growth,
compare the two graphs below. The California’s population and the
plotted points on the graph of y=5.68(1.32)x. We picked 1.32 because it is
an average growth factor for California for the six decades. Except for
the differences in the scale on the horizontal axis, they look quite a bit
alike.
Population of California
35000000
Population
30000000
25000000
20000000
Series1
15000000
Series2
10000000
5000000
0
1
2
3
4
5
6
7
Year 1930-1990
y=5.68(1.32)x
35
30
25
20
Series1
15
10
5
0
1
2
3
4
5
6
Reference:
McConnell, J,W. and others (1998). Algebra (Integrated Mathematics),
The University of Chicago School of Mathematics Project, Scott,
Foresman and Company, Glenview, Illinois