To answer the questions about the rat population in a major metropolitan city, we will use the given formula [tex]\( n(t) = 82 e^{0.015 t} \)[/tex], where [tex]\( t \)[/tex] is measured in years since 2001, and [tex]\( n(t) \)[/tex] is measured in millions.
### Part 1: Rat Population in 2001
To find the rat population in 2001, we need to evaluate the formula at [tex]\( t = 0 \)[/tex] (since [tex]\( t \)[/tex] represents the number of years since 2001).
[tex]\[
n(0) = 82 e^{0.015 \cdot 0}
\][/tex]
Since [tex]\( 0.015 \cdot 0 = 0 \)[/tex], the equation simplifies to:
[tex]\[
n(0) = 82 e^0
\][/tex]
We know that [tex]\( e^0 = 1 \)[/tex], thus:
[tex]\[
n(0) = 82 \times 1 = 82
\][/tex]
So, the rat population in 2001 was:
[tex]\[
82 \text{ million rats}
\][/tex]
### Part 2: Rat Population in 2019
To find the predicted rat population in 2019, we need to evaluate the formula at [tex]\( t = 2019 - 2001 = 18 \)[/tex].
[tex]\[
n(18) = 82 e^{0.015 \cdot 18}
\][/tex]
Calculating the exponent:
[tex]\[
0.015 \cdot 18 = 0.27
\][/tex]
So, the equation becomes:
[tex]\[
n(18) = 82 e^{0.27}
\][/tex]
From the result given, we know that the exponential calculation and multiplication result in:
[tex]\[
n(18) = 107.41708496012627
\][/tex]
So, the model predicts that the rat population in 2019 was:
[tex]\[
107.42 \text{ million rats} \text{ (rounded to two decimal places)}
\][/tex]
Therefore, the solutions to the questions are:
1. The rat population in 2001 was [tex]\( 82 \)[/tex] million rats.
2. The model predicts that the rat population in 2019 was approximately [tex]\( 107.42 \)[/tex] million rats.
