Question

The population of a city after t years is given by P(t) = 13,518e^0.041t, where t O corresponds to the current year.How many years from the current year will it take for the population of the city to reach 55,000?Round to the nearest hundredth of a year.In approximatelyyears the population of the city will reach 55,000.

Answer 1

1) Gathering the data

$\begin{gathered} P(t)=\text{ 13,158 }\times10^{0.041t} \\ t_0=Current\text{ year} \\ \end{gathered}$

2) To find how many years from the current year will it take, let's plug into that equation:

$\begin{gathered} P(t)=\text{ 13,158 }\times e^{0.041t} \\ 55,000=13158\times e^{0.041t} \\ \frac{55000}{13158}=\frac{13518}{13518}\times e^{0.041t}^{} \\ 4.1799=e^{0.041t}^{} \\ \ln (4.1799)\text{ =}\ln (e^{0.041t}) \\ 1.4302=0.041t \\ t=34.88\cong35 \end{gathered}$

3) So approximately 35 years from the current year.