Question

Some time in the future a human colony is started on Mars. The colony begins with 50000 people and grows exponentially to 200000 in 200 years. Give a formula for the size of the human population on Mars as a function of t

Answer 1

Answer:

$P(t) = 50000 e^{0.0069315 t}$

Step-by-step explanation:

For this case since the population follows an exponential model we have the general equation:

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

Where P(t) represent the population at time t. t represent the years since the starting year.

r represent the growth/dcay constant rate

For this case we have the initial condition given : $P(0) = 50000$ and if we replace this into the general equation we have:

$50000 = P_o e^{r*0} = P_o$

And the equation would be:

$P(t) = 50000 e^{rt}$

Now we can use the second condition given $P(200) =200000$ and replacing into the general formula we got:

$200000= 50000 e^{200t}$

We can divide both sides by 50000 and we got:

$4 = e^{200t}$

Now we can apply natural log on both sides:

$ln(4) = 200t$

And then:

$t = \frac{ln(4)}{200}=0.0069315$

So then our final equation would be given by:

$P(t) = 50000 e^{0.0069315 t}$