Question

the town of gilbertville increased from a population of 3562 people in 1970 to a population of 9765 in 2000. define an exponential function that models the towns population as a function of the number of years since 1970

Answer 1

P(t) = 3562•( 1.034)^t

Explanation:

Here, we want to define an exponential equation.

To define the exponential equation, some key concepts are needed.

We need the initial population value which is given as; 3562

We need the growth rate per year , let’s call this r

The population at a particular year can thus be modeled as;

P(t) = I•(1 + r)^t

where t is the number of years after 1970( difference between the particular year and 1970)

Thus, to completely write the exponential equation, we need to get the value of r ( which is the growth percentage per year assuming we have a parallel growth rate of the population).

Hence;

9765 = 3562( 1 + r)^30

divide both sides by 3562

2.7414 = (1 + r)^30

Take the log to base e of both sides

ln 2.7414 = ln (1 + r)^30

ln 2.7414 = 30ln (1 + r)

1.0085 = 30 ln (1 + r)

divide both sides by 30

0.034 = ln ( 1 + r)

1 + r = e^(0.034)

1 + r = 1.034

r = 1.034 - 1

r = 0.034

So the growth percentage is about 3.4% yearly

So the exponential equation would be;

P(t) = 3562•( 1.034)^t