The rat population in a major metropolitan city is given by the formula $n(t)=27 e^{0.04 t}$ where $t$ is measured in years since 1991 and $n$ is measured in millions. (a) What was the rat population in 1991? (b) What is the rat population going to be in the year 2001?
Solution
Step 1 ：The rat population in a major metropolitan city is given by the formula \(n(t)=27 e^{0.04 t}\) where \(t\) is measured in years since 1991 and \(n\) is measured in millions.
Step 2 ：For part (a), we need to find the rat population in 1991. According to the problem, \(t\) is measured in years since 1991. So, for the year 1991, \(t=0\). We need to substitute \(t=0\) into the formula and calculate the population.
Step 3 ：For part (b), we need to find the rat population in 2001. According to the problem, \(t\) is measured in years since 1991. So, for the year 2001, \(t=2001-1991=10\). We need to substitute \(t=10\) into the formula and calculate the population.
Step 4 ：Substituting the values of \(t\) into the formula, we get the rat population in 1991 as \(27.0\) million and the rat population in 2001 as approximately \(40.2792668363143\) million.
Step 5 ：Final Answer: The rat population in 1991 was \(\boxed{27}\) million. The rat population in 2001 was approximately \(\boxed{40.28}\) million.
