Question

Need help with b please and thank you (lots of points)

Answer 1

17.34 years

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.

`P(t)=(10000)/(1+11.5e^(-0.1408614477t))`

Setting this equal to 5000 and multiplying by the denominator, we have ...

`5000=(10000)/(1+11.5e^(-0.1408614477t))\\\\5000+57500e^(-0.1408614477t)=10000\\\\e^(-0.1408614477t)=(5000)/(57500)\qquad\text{subtract 5000, divide by 57500}\\\\-0.1408614477t = \ln{(1)/(11.5)}=-\ln(11.5)\qquad\text{take logs}\\\\t=(\ln(11.5))/(0.1408614477)\approx17.3386`

For the population to reach 5000, it will take about 17.34 years.

_____

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.