Question

The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 25% in 10 years. What will be the population in 40 years

Answer 1

Answer:

The population in 40 years will be 1220.

Step-by-step explanation:

The population of a town grows at a rate proportional to the population present at time t.

This means that:

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

In which P(t) is the population after t years, P(0) is the initial population and r is the growth rate.

The initial population of 500 increases by 25% in 10 years.

This means that $P(0) = 500, P(10) = 1.25*500 = 625$

We apply this to the equation and find t.

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

$625 = 500e^{10r}$

$e^{10r} = \frac{625}{500}$

$e^{10r} = 1.25$

Applying ln to both sides

$\ln{e^{10r}} = \ln{1.25}$

$10r = \ln{1.25}$

$r = \frac{\ln{1.25}}{10}$

$r = 0.0223$

So

$P(t) = 500e^{0.0223t}$

What will be the population in 40 years

This is P(40).

$P(t) = 500e^{0.0223t}$

$P(40) = 500e^{0.0223*40} = 1220$

The population in 40 years will be 1220.