To solve the problem of determining when the environment can no longer support the population given the exponential growth of the organisms, we can break it down step-by-step:
### Step 1: Finding the Growth Rate
From the given table:
[tex]\[
\begin{array}{|c|c|}
\hline
y \text{ years} & \text{number of organisms}, n \\
\hline
1 & 55 \\
\hline
2 & 60 \\
\hline
3 & 67 \\
\hline
4 & 75 \\
\hline
\end{array}
\][/tex]
First, calculate the growth rate for each year:
[tex]\[
\text{Growth rate from year 1 to 2} = \frac{60}{55} \approx 1.09091
\][/tex]
[tex]\[
\text{Growth rate from year 2 to 3} = \frac{67}{60} \approx 1.11667
\][/tex]
[tex]\[
\text{Growth rate from year 3 to 4} = \frac{75}{67} \approx 1.11940
\][/tex]
### Step 2: Average Growth Rate
Calculate the average growth rate:
[tex]\[
\text{Average growth rate} = \frac{1.09091 + 1.11667 + 1.11940}{3} \approx 1.10899
\][/tex]
### Step 3: Project the Population
Starting from year 4 with 75 organisms and continuing with an average growth rate of approximately 1.10899, we project the population each year until it exceeds 600 organisms.
### Step 4: Calculating the Year
Using the exponential growth formula [tex]\( P_t = P_0 \times r^t \)[/tex]:
- [tex]\( P_0 = 75 \)[/tex] (initial population at year 4)
- [tex]\( r = 1.10899 \)[/tex] (average growth rate)
We want to find [tex]\( t \)[/tex] when [tex]\( P_t \)[/tex] (the population) exceeds 600.
Through iteration (since each year is calculated by multiplying the previous year's population by the growth rate until exceeding 600), we find:
After 24 years, the population grows beyond the environmental limit of 600.
### Conclusion
After 25 years in total:
[tex]\[
\text{The environment will no longer be able to support the population.}
\][/tex]
Thus, after how many years will the environment no longer be able to support the population?
The answer is 25 years.
