To determine the best decision based on the zebra mussel population in Lake Wilson, we first need to create an exponential regression model from the given data points. The provided data is as follows:
- Year (t): 0, 2, 4, 6, 8
- Population (P): 1000, 2200, 4800, 10560, 23100
We can model the population using an exponential function of the form:
\[ P(t) = P_0 \cdot e^{kt} \]
Where:
- \( P_0 \) is the initial population at year 0.
- \( k \) is the growth rate.
- \( t \) is the time in years.
Step 1: Calculate Initial Population and Growth Rate
Given:
- \( P_0 = 1000 \) (initial population at year 0)
- The population at different years is as follows:
- \( P(2) = 2200 \)
- \( P(4) = 4800 \)
- \( P(6) = 10560 \)
- \( P(8) = 23100 \)
Using one of the known points to solve for \( k \):
Using the point at year 2: \[ 2200 = 1000 \cdot e^{2k} \] \[ 2.2 = e^{2k} \] Taking the natural log on both sides: \[ \ln(2.2) = 2k \] \[ k = \frac{\ln(2.2)}{2} \approx 0.396 \]
Now, we can use \( k \) to estimate the population for year 10:
Step 2: Estimate Population at Year 10
Using the exponential model: \[ P(10) = 1000 \cdot e^{10k} \] With \( k \approx 0.396 \): \[ P(10) = 1000 \cdot e^{10 \cdot 0.396} \] \[ P(10) = 1000 \cdot e^{3.96} \] Calculating \( e^{3.96} \): \[ e^{3.96} \approx 52.9 \] Therefore: \[ P(10) = 1000 \cdot 52.9 \] \[ P(10) \approx 52900 \]
Step 3: Analyze Results
Now that we have \( P(10) \approx 52900 \), we can make a decision based on the given thresholds:
- If the population exceeds 50,000, implement control measures immediately (threshold option 2).
- If the population is between 40,000 and 50,000, control measures should be implemented over the next 2 years (threshold option 3).
- If below 40,000, wait (threshold option 1).
Since the estimated population is approximately 52,900, which exceeds 50,000, the correct choice is:
Answer: b. implement the control measures immediately as the model predicts the population to exceed 50,000 by year 10.