To solve this problem, we need to establish an exponential growth model for the population of 15,000 organisms, which grows at a rate of 5.2% each year.
Let's break this down step by step.
### Step 1: Determine the initial population
The initial population, [tex]\( P_{\text{initial}} \)[/tex], is given as 15,000 organisms. Therefore, in our model:
[tex]\[ a = 15000 \][/tex]
### Step 2: Convert the growth rate to a decimal
The annual growth rate is given as 5.2%. To work with this rate in our exponential model, we need to convert it to a decimal. The percentage rate of 5.2% can be converted as follows:
[tex]\[ \text{Growth rate} = \frac{5.2}{100} = 0.052 \][/tex]
### Step 3: Determine the growth factor
The growth factor, [tex]\( b \)[/tex], reflects how much the population increases each year. It is calculated as one plus the growth rate:
[tex]\[ b = 1 + \text{Growth rate} = 1 + 0.052 = 1.052 \][/tex]
### Step 4: Write the exponential model
We are supposed to express the population [tex]\( P \)[/tex] as a function of time [tex]\( t \)[/tex]. The general form of the exponential growth model is:
[tex]\[ P = a \cdot b^t \][/tex]
Substituting in the values we have determined:
[tex]\[ P = 15000 \cdot 1.052^t \][/tex]
### Final Model
Thus, an exponential model for the population [tex]\( P \)[/tex] as a function of time [tex]\( t \)[/tex] in years is:
[tex]\[ P = 15000 \cdot (1.052)^t \][/tex]
This model describes how the population of organisms will grow over time, taking into account the initial population and the given annual growth rate.
