Question

The population of a region is growing exponentially. There were 40 million people in 1980 (when ????=0) and 80 million people in 1990. Find an exponential model for the population (in millions of people) at any time ????, in years after 1980.

Answer 1

Answer:

$P(t) = 40e^{0.06931t}$

Step-by-step explanation:

The population exponential equation is as follows.

$P(t) = P(0)e^{rt}$

In which P(t) is the population in t years from now, P(0) is the population in the current year and r(decimal) is the growth rate.e = 2.71 is the Euler number.

Find an exponential model for the population (in millions of people) at any time ????, in years after 1980.

There were 40 million people in 1980 (when ????=0).

This means tht P(0) = 40.

So

$P(t) = P(0)e^{rt}$

$P(t) = 40e^{rt}$

80 million people in 1990.

1990 is 10 years after 1980. So P(10) = 80. We use this to find the value of r.

So

$P(t) = 40e^{rt}$

$80 = 40e^{10r}$

$e^{10r} = 2$

Applying ln to both sides, since $\ln{e^{a}} = a$

$\ln{e^{10r}} = \ln{2}$

$10r = 0.6931$

$r = \frac{0.6931}{10}$

$r = 0.06931$

So the exponential model for the population is:

$P(t) = 40e^{0.06931t}$