To determine which year the squirrel population reached the carrying capacity for its particular forest area, we'll follow a step-by-step approach based on the logistic growth model. This model is used to describe growth that is initially exponential, but slows as the population approaches the carrying capacity due to environmental limitations.
### Step-by-Step Solution:
1. Understanding the Logistic Growth Model:
- The logistic growth equation is given by:
[tex]\[
P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right) e^{-rt}}
\][/tex]
Where:
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex]
- [tex]\( K \)[/tex] is the carrying capacity
- [tex]\( P_0 \)[/tex] is the initial population
- [tex]\( r \)[/tex] is the growth rate
- [tex]\( t \)[/tex] is the time in years
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828)
2. Given Values:
- Initial population, [tex]\( P_0 = 100 \)[/tex] squirrels
- Carrying capacity, [tex]\( K = 1000 \)[/tex] squirrels
- Growth rate, [tex]\( r = 0.3 \)[/tex] (or 30%)
- Duration, [tex]\( t = 10 \)[/tex] years
3. Calculation:
- We calculate the population each year up to 10 years using the logistic growth equation.
4. Determining the Reached Capacity Year:
- We need to find the year [tex]\( t \)[/tex] where [tex]\( P(t) \)[/tex] becomes equal to or exceeds [tex]\( K \)[/tex].
### Result after Calculation:
[tex]\[ \text{Population after 10 years} = 999.9993856960735 \text{ squirrels} \][/tex]
### Conclusion:
Based on the model and calculations, the population of squirrels after 10 years is approximately 1000, but it has not exceeded the carrying capacity within the observed timeframe. Therefore, the population does not exactly hit the carrying capacity [tex]\( K = 1000 \)[/tex] squirrels at any distinct year before reaching year 11. Thus:
- The squirrel population did not reach the carrying capacity within the 10 years.
- The year when this might happen, if it ever did, is still pending a more extended timeframe beyond the given 10 years.
In summary, by analyzing the logistic growth model, we conclude that the population did not strictly reach the carrying capacity within the first 10 years.
