Answer:
The growth rate of the town is approximately 0.212 per month.
Explanation:
The exponential model is defined by the following formula:
[tex]n = n_{o}\cdot e^{r\cdot t}[/tex] (1)
Where:
[tex]n_{o}[/tex] - Initial population, dimensionless.
[tex]r[/tex] - Growth rate, measured in [tex]\frac{1}{mo}[/tex].
[tex]t[/tex] - Time, measured in months.
[tex]n[/tex] - Current population, dimensionless.
From (1), we clear the growth rate as follows:
[tex]\frac{n}{n_{o}} = e^{r\cdot t}[/tex]
[tex]r = \frac{1}{t}\cdot \ln \frac{n}{n_{o}}[/tex] (2)
If we know that [tex]n_{o} = 45[/tex], [tex]n = 360[/tex] and [tex]t = 15\,mo[/tex], then the growth rate is:
[tex]r = \left(\frac{1}{15\,mo}\right)\cdot \ln \frac{360}{15}[/tex]
[tex]r \approx 0.212\,\frac{1}{mo}[/tex]
The growth rate of the town is approximately 0.212 per month.
Using the exponential function, the growth rate experienced during the California's gold rush is [tex] 13.86% [/tex]
The exponential function :
Substituting the values into the equation :
[tex] 360 = 45e^{15r} [/tex]
Divide both sides by 45
[tex] \frac{360}{45}= e^{15r} [/tex]
[tex] 8 = e^{15r} [/tex]
Take the In of both sides
[tex] 2.07944 = 15r [/tex]
[tex] r = \frac{2.07944}{15} = 0.13863 [/tex]
Hence, the monthly growth rate is 13.86%
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