Answered

A population numbers 18,000 organisms initially and decreases by [tex]$3.5 \%$[/tex] each year.

Suppose [tex]$P$[/tex] represents the population, and [tex]$t$[/tex] the number of years of growth. An exponential model for the population can be written in the form [tex]$P = a \cdot b^t$[/tex], where

[tex]$
P = \square
$[/tex]



Answer :

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].