Jump to level 1

The population of a city is modeled by the function $P(t)=8000 e^{0.1622 t}$. What is the population of the town after 6 years? Ex: 12345 people After how many years will the population be 41000 ? Ex: 1.234 years

Step 1 ：The population of a city is modeled by the function \(P(t)=8000 e^{0.1622 t}\).

Step 2 ：We are asked to find the population of the town after 6 years. This can be calculated by substitifying \(t=6\) in the given function.

Step 3 ：By substituting \(t=6\) into the function, we get \(P(6)=8000 e^{0.1622 \times 6}\) which is approximately 21171.195218543555.

Step 4 ：So, the population of the town after 6 years is approximately \(\boxed{21171}\) people.

Step 5 ：We are also asked to find out after how many years the population will be 41000. This can be calculated by setting \(P(t) = 41000\) and solving for \(t\).

Step 6 ：By setting \(P(t) = 41000\) and solving for \(t\), we get \(t = \frac{\ln(\frac{41000}{8000})}{0.1622}\) which is approximately 10.074787453911664.

Step 7 ：So, the population will be 41000 after approximately \(\boxed{10.07}\) years.