Question

The population of a town doubled approximately every 5 years during the first several decades after it was founded. If the original population of the town was 39 people when it was founded in the year 1761, about how many residents were there in the year 1781? (Hint: the population doubled 4 times during this time period.)

Answer 1

624 people

(I put two ways to look at the problem.)

Explanation:

What is describe here is an exponential function of the form:

`P=P_0 e^(kt)`

`t` is the number of years after 1761.

`P_0` is the initial population.

So t=0 represents year 1761.

t=1 represents year 1762

t=2 represents year 1763

....

t=20 represents year 1781.

So we have the doubling time is 5 years. This means the population will be twice what it was in 5 years. Let's plug this into:

`P=P_0e^(kt)`

`2P_0=P_0e^(k\cdot 5)`

Divide both sides by
`P_0`:

`2=e^(5k)`

Convert to logarithm form:

`5k=\ln(2)`

Multiply both sides by 1/5:

`k=(1)/(5)\ln(2)`

`k=\ln(2^{(1)/(5)})` By power rule.

So in the next sentence they actually give us the initial population and we just found k so this is our function for P:

`P=39e^{\ln(2^{(1)/(5)})t}`

So now we plug in 20 to find how many residents there were in 1761:

`P=39e^{\ln(2^{(1)/(5)})(20)}`

This is surely going to the calculator:

`P=624`

Now if you don't like that, let's try this:

Year 0 we have 39 people.

Year 5 we have 39(2)=78 people.

Year 10 we have 78(2)=156 people.

Year 15 we have 156(2)=312 people.

Year 20 we have 312(2)=624 people.