To model the population decrease over time, we need to use the exponential decay formula, which is given by:
[tex]\[ P = a \cdot b^t \][/tex]
where:
- [tex]\( P \)[/tex] is the population at time [tex]\( t \)[/tex].
- [tex]\( a \)[/tex] is the initial population.
- [tex]\( b \)[/tex] is the base of the exponential, which represents the factor by which the population decreases each year.
- [tex]\( t \)[/tex] is the time in years.
Let's go through the steps to determine the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
1. Initial Population ([tex]\( a \)[/tex]): The problem states that the initial population is 16,000 organisms. Therefore,
[tex]\[ a = 16000 \][/tex]
2. Annual Decrease Rate: The population decreases by 8.9% each year. To find [tex]\( b \)[/tex], we need to understand that the population retains a percentage of its value after each year. If it decreases by 8.9%, it retains [tex]\( 100\% - 8.9\% = 91.1\% \)[/tex] of its population each year.
To express this as a decimal:
[tex]\[ \text{Retained population} = 91.1\% = 0.911 \][/tex]
Therefore,
[tex]\[ b = 0.911 \][/tex]
Combining these values into our exponential model, we have:
[tex]\[ P = 16000 \cdot 0.911^t \][/tex]
