Absolutely, let's tackle this step-by-step.
To determine the exponential model for the population [tex]$P$[/tex], let's identify the values for [tex]$a$[/tex] and [tex]$b$[/tex] in the equation [tex]\( P = a \cdot b^t \)[/tex].
### Step-by-Step Solution:
1. Initial Population ([tex]$a$[/tex]):
The population starts at 18,000 organisms. This is the initial value when [tex]\( t = 0 \)[/tex]. Therefore,
[tex]\[
a = 18000
\][/tex]
2. Annual Decrease Rate ([tex]$\%)$[/tex]:
The population decreases by [tex]\( 3.5\% \)[/tex] each year. We need to convert this percentage into a decay factor.
3. Determining the Decay Factor ([tex]$b$[/tex]):
A decrease by [tex]\( 3.5\% \)[/tex] can be interpreted as the population retaining [tex]\( 100\% - 3.5\% = 96.5\% \)[/tex] of its value each year. The decay factor is:
[tex]\[
b = 0.965
\][/tex]
### Constructing the Exponential Model:
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the model [tex]\( P = a \cdot b^t \)[/tex], we get:
[tex]\[
P = 18000 \cdot 0.965^t
\][/tex]
So, the completed exponential model for the population is:
[tex]\[
P = 18000 \cdot 0.965^t
\][/tex]
This model describes how the population of organisms decreases over time at an annual rate of [tex]\( 3.5\% \)[/tex].
