MATH 120 Lecture Notes - Lecture 28: Ibm 7090
Here is an exponential modeling example.
In 1980, City X had a population of 1.2 million. City X’s population increases by 0.8% every
year.
In 1990, City Y had twice as many people as City X, but only 85% more people than City X
in 2005.
If City Y’s population is growing exponentially, when will the populations be equal?
We begin by finding exponential models for each city. We will express these in units of millions.
Let t= 0 represent 1980, so that throughout trepresents years after 1980.
Let PXbe the population of City X. Then we seek A0and b, constants, such that
PX=A0bt.
Since PX= 1.2when t= 0, we have A0= 1.2.
Then since the population increases by 0.8% per year, we know that b= 1.008.
In general, a constant increase of r%per year translates to a bvalue of 1 + b
100 .
Thus
PX= 1.2(1.008)t.
(Since 1.008 = eln1.008 =e0.0079681696, we may write
PX= 1.2e0.0079681696t= 1.2e0.0079681696t
which is of the form A0est, another popular form for exponential functions.)
For city Y, we first calculate the population of City X in 1990. We have
PX= 1.2(1.008)10 = 1.2995.
In 1990, City Y’s population was twice this, i.e., PY= 2.5991.
In 2005, City X had a population of 1.2(1.008)25 = 1.4645. City Y was 85% more than this, so
PY= 1.85(1.4645) = 2.7094.
This gives us two data points for City Y. Assuming PY=B0ct, we seek B0and c, from the
equations
2.5991 = B0c10
2.7094 = B0c25
Dividing the first into the second, we find
2.7094
2.5991 = 1.04244 = c15
so c= 1.042441/15 = 1.0027746.
Document Summary
In 1980, city x had a population of 1. 2 million. City x"s population increases by 0. 8% every year. In 1990, city y had twice as many people as city x, but only 85% more people than city x in 2005. We begin by nding exponential models for each city. We will express these in units of millions. Let t = 0 represent 1980, so that throughout t represents years after 1980. Let px be the population of city x. Then we seek a0 and b, constants, such that. Since px = 1. 2 when t = 0, we have a0 = 1. 2. Then since the population increases by 0. 8% per year, we know that b = 1. 008. In general, a constant increase of r% per year translates to a b value of 1 + b. Px = 1. 2(1. 008)t. (since 1. 008 = eln1. 008 = e0. 0079681696, we may write. = 1. 2e0. 0079681696t which is of the form a0est, another popular form for exponential functions. )
