Answer:
17.34 years
Step-by-step explanation:
The equation you wrote in the first part of the problem can be solved for the value of t that makes the population be 5000.
[tex]P(t)=\dfrac{10000}{1+11.5e^{-0.1408614477t}}[/tex]
Setting this equal to 5000 and multiplying by the denominator, we have ...
[tex]5000=\dfrac{10000}{1+11.5e^{-0.1408614477t}}\\\\5000+57500e^{-0.1408614477t}=10000\\\\e^{-0.1408614477t}=\dfrac{5000}{57500}\qquad\text{subtract 5000, divide by 57500}\\\\-0.1408614477t = \ln{\dfrac{1}{11.5}}=-\ln(11.5)\qquad\text{take logs}\\\\t=\dfrac{\ln(11.5)}{0.1408614477}\approx17.3386[/tex]
For the population to reach 5000, it will take about 17.34 years.
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Additional comment
The value -0.14086... is the natural log of the ratio 1818/2093. This means the "exact answer" is ln(11.5)/(ln(2093) -ln(1818)), an irrational number.
A graphing calculator can answer the question easily.
