Question

A town has a population of 2093 people at time t=0.The population increases by 42 people a year. Enter a formula for the population, P, of the town as a function of year t.

Answer 1

P(t) = 2093e^(42t).

Explanation:

The population of this town can be modeled by the following differential equation

dP/dt = Pr

where r is the growth rate in people a year.

We can solve this differential equation by the separation of variables method.

dP/P = rdt

Integrating both sides, we have:

ln P = rt + P0

where P0 is the initial population

To isolate P, we do this:

e^(ln P) = e^(rt + P0)

P(t) = P0e^(rt).

We have that the population increases by 42 people a year, so r = 42. We also have that the population at time t = 0 is 2093 people, so P0 = 2093.

So the formula for the population, P, of the town as a function of year t is P(t) = 2093e^(42t).