Let's solve the problem step by step.
### Step 1: Identify the Variables
We need to determine the initial population, the daily growth rate, and the number of days. These are the variables in our exponential growth formula \( f(t) = P e^{r t} \) where:
- \( P \) is the initial population,
- \( r \) is the growth rate (expressed as a decimal),
- \( t \) is the time period in days.
From the problem statement, we have:
[tex]\[ P = 16 \text{ (initial population)} \][/tex]
[tex]\[ r = 3.6\% = 0.036 \text{ (daily growth rate in decimal form)} \][/tex]
[tex]\[ t = 30 \text{ (number of days)} \][/tex]
### Step 2: Apply the Exponential Growth Formula
We use the exponential growth formula to find the population after \( t \) days:
[tex]\[ f(t) = P \cdot e^{r \cdot t} \][/tex]
Let's substitute the known values into the formula:
[tex]\[ f(30) = 16 \cdot e^{0.036 \cdot 30} \][/tex]
### Step 3: Calculate the Population
According to the calculations:
[tex]\[ P = 16 \][/tex]
[tex]\[ r = 0.036 \][/tex]
[tex]\[ t = 30 \][/tex]
After performing the necessary calculations, the result is approximately:
[tex]\[ f(30) = 16 \cdot e^{1.08} \][/tex]
[tex]\[ f(30) \approx 47.11 \][/tex]
Thus, the population after 30 days is approximately 47.11 organisms.
### Conclusion
After 30 days, the number of organisms in the petri dish will be approximately 47.11. This is calculated using the exponential growth model with an initial population of 16 organisms and a daily growth rate of 3.6%.
