Respuesta :
Answer:
624 people
(I put two ways to look at the problem.)
Step-by-step explanation:
What is describe here is an exponential function of the form:
[tex]P=P_0 e^{kt}[/tex]
[tex]t[/tex] is the number of years after 1761.
[tex]P_0[/tex] 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:
[tex]P=P_0e^{kt}[/tex]
[tex]2P_0=P_0e^{k\cdot 5}[/tex]
Divide both sides by [tex]P_0[/tex]:
[tex]2=e^{5k}[/tex]
Convert to logarithm form:
[tex]5k=\ln(2)[/tex]
Multiply both sides by 1/5:
[tex]k=\frac{1}{5}\ln(2)[/tex]
[tex]k=\ln(2^{\frac{1}{5}})[/tex] 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:
[tex]P=39e^{\ln(2^{\frac{1}{5}})t}[/tex]
So now we plug in 20 to find how many residents there were in 1761:
[tex]P=39e^{\ln(2^{\frac{1}{5}})(20)}[/tex]
This is surely going to the calculator:
[tex]P=624[/tex]
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.
