Question

The population of a town in increasing linearly. 14-In 2016, the population was 14,200, and the popula-tion grew to 17,500 in 2019. If this trend continues,in what year will the population exceed 20,000?

Answer 1

So we know that the population of a town grows linearly through the years. This means that the equation that relates the town's population (P) with the years (t) is a linear one and can be written as:

$P=mt+b$

In order to make the calculations simpler I'm going to take 2016 as the year 0 so 2019 would actually be t=3. Now we take the data given by the statement of the question to build two equations replacing P and t with the corresponding value in each case:

$\begin{gathered} 14200=m\cdot0+b=b \\ 17500=m\cdot3+b \end{gathered}$

So from the first equation we have b=14200. If we use this in the second equation we get:

$17500=3m+14200$

We can substract 14200 from both sides of this equation:

$\begin{gathered} 17500-14200=3m+14200-14200 \\ 3300=3m \end{gathered}$

And we divide both sides by 3:

$\begin{gathered} \frac{3300}{3}=\frac{3m}{3} \\ m=1100 \end{gathered}$

Then the equation that relates the town's population and the years is:

$P=1100\cdot t+14200$

If we want to know the year when the population reaches 20000 we have to take P=20000 and solve for t:

$20000=1100t+14200$

We substract 14200 from both sides:

$\begin{gathered} 20000-14200=1100t+14200-14200 \\ 5800=1100t \end{gathered}$

And we divide both sides by 1100:

$\begin{gathered} \frac{5800}{1100}=\frac{1100t}{1100} \\ t=5.27 \end{gathered}$

So the population reaches 20000 5.27 years after 2016 which means that in 2021 the population exceeds 2