Question

Based on the census data, the population of whoville was 43,230 in the year of 2000 and has been growing at a rate of 3.5%

Answer 1

The model would take the form of

$P(t)=P(0)e^{0.035t}$

So in 2015, when $t=15$, the population would be

$P(15)=43230e^{0.035\times15}\approx73079$

On the other hand, you could use the following discrete model for the population, which gives a similar result. If $t=0$ corresponds to the year 2000, then in 2001, when $t=1$, you have

$P(1)=P(0)\times1.035^1\approx44743$

In the next year,

$P(2)=P(1)\times1.035^1=P(0)\times1.035^2\approx46309$

And so on, making the general pattern for the $t$th year to be

$P(t)=P(t-1)\times1.035^1=P(t-2)\times1.035^2=\cdots=P(0)\times1.035^t$

This means in the year 2015, or when $t=15$, the population should be approximately

$P(15)=P(0)\times1.035^{15}\approx72425$