Johnny's assumption is that, none of the residents of the town will live up to 231 years.
The parameters are given as:
[tex]\mathbf{P_o = 10000}[/tex]--- the population in 2005
[tex]\mathbf{r = 0.3\%}[/tex] -- the rate at which the population declines
To model a population, we make use of the following exponential function
[tex]\mathbf{P = P_o \times(1 -r)^t}[/tex]
Where P is the current population in t years.
So, we have:
[tex]\mathbf{P = 10000 \times(1 -0.3\%)^t}[/tex]
When the population reaches half of the initial population, then P = 5000.
So, we have:
[tex]\mathbf{5000 = 10000 \times(1 -0.3\%)^t}[/tex]
Divide both sides by 10000
[tex]\mathbf{0.5 = (1 -0.3\%)^t}[/tex]
[tex]\mathbf{0.5 = (0.997)^t}[/tex]
Take logarithm of both sides
[tex]\mathbf{log(0.5) = log(0.997)^t}[/tex]
Rewrite the equation as
[tex]\mathbf{log(0.5) = tlog(0.997)}[/tex]
Make t the subject
[tex]\mathbf{t = \frac{log(0.5)}{log(0.997)}}[/tex]
[tex]\mathbf{t = 231}[/tex]
This means that the population will halve its initial population in 231 years.
Johnny's assumption is that, none of the residents of the town will live up to 231 years.
Read more about population functions at:
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