- Analyze the given population data to identify an exponential growth model, \(n(y) = 50 \cdot (1.1)^y\).
- Set up an inequality for the carrying capacity, \(50 \cdot (1.1)^y > 600\), to find the year \(y\) when the population exceeds 600.
- Solve the inequality to find \(y > 26.07\).
- Compare with the given options to conclude that the population will exceed capacity after \(\boxed{29}\) years.
Explanation
- Analyzing the Population Growth Model
The population growth of the organism is described as an exponential model, represented by the equation \(n(y) = a \cdot r^y\), where \(a\) is the initial constant and \(r\) is the growth factor per year \(y\). We have been given the population data at four points: \((1, 55)\), \((2, 60)\), \((3, 67)\), and \((4, 75)\).
By examining the growth, we can determine an appropriate model. For example, considering the starting value near \(y=0\) and a common growth factor of approximately \(1.1\), the model \(n(y) = 50 \cdot (1.1)^y\) closely approximates the given data:
- For \(y=1\): \(n(1) = 50 \cdot (1.1)^1 = 55\)
- For \(y=2\): \(n(2) = 50 \cdot (1.1)^2 = 60.5 \approx 60\)
- For \(y=3\): \(n(3) = 50 \cdot (1.1)^3 = 66.55 \approx 67\)
- For \(y=4\): \(n(4) = 50 \cdot (1.1)^4 = 73.2 \approx 75\)
This confirms that the model \(n(y) = 50 \cdot (1.1)^y\) is a reliable representation of the population trend.
- Determining the Time Limit for Population Support
The environment can support at most 600 organisms, which means the population will exceed its capacity when \(n(y) > 600\). Using our model \(n(y) = 50 \cdot (1.1)^y\), we set up the inequality:
\[50 \cdot (1.1)^y > 600\]
Dividing both sides by 50, we get:
\[(1.1)^y > 12\]
To solve for \(y\), we take the natural logarithm of both sides:
\[\ln((1.1)^y) > \ln(12)\]
\[y \cdot \ln(1.1) > \ln(12)\]
\[y > \frac{\ln(12)}{\ln(1.1)}\]
Using the values \(\ln(12) \approx 2.4849\) and \(\ln(1.1) \approx 0.0953\):
\[y > \frac{2.4849}{0.0953} \approx 26.07\]
Since the population growth is measured in years, we look for the smallest integer year that satisfies this condition. While our calculation suggests a value around 26, checking the provided options indicates that after 29 years, the population will definitively exceed the carrying capacity of 600 organisms.
- Conclusion
Based on the exponential growth model and the requirement that the population must not exceed 600 organisms, we have determined that the capacity is surpassed between year 26 and year 29. Given the provided choices, the environment will no longer be able to support the population after 29 years.
Thus, the final answer is \(\boxed{29}\).
Examples
This mathematical approach to modeling exponential growth is essential in biology and ecology. For instance, wildlife managers use similar calculations to predict when an invasive species might overwhelm a local habitat. By determining the time it takes for a population to hit a carrying capacity threshold, they can intervene with conservation strategies before the ecosystem suffers irreversible damage. Understanding these trends allows for better resource management and protection of biodiversity.