1. **Problem statement:** We have a fish population of 50000 that decreases by 9% annually. We want to find the population after $t$ years. 2. **Formula and explanation:** The exponential decay formula is: $$N(t) = N_0 \cdot (1 - r)^t$$ where $N_0$ is the initial population, $r$ is the decay rate (as a decimal), and $t$ is time in years. 3. **Apply values:** Here, $N_0 = 50000$, $r = 0.09$, so: $$N(t) = 50000 \cdot (1 - 0.09)^t = 50000 \cdot 0.91^t$$ 4. **Interpretation:** This formula gives the fish population after $t$ years considering a 9% annual decrease. --- 1. **Problem statement:** Authorities want to act when the population has decreased by 10000 fish. 2. **Set up equation:** The population after $t$ years is $N(t) = 50000 \cdot 0.91^t$. They want to find $t$ such that: $$50000 - N(t) = 10000$$ 3. **Solve for $t$:** $$50000 - 50000 \cdot 0.91^t = 10000$$ $$50000(1 - 0.91^t) = 10000$$ $$1 - 0.91^t = \frac{10000}{50000} = 0.2$$ $$0.91^t = 0.8$$ 4. **Take natural logarithm:** $$\ln(0.91^t) = \ln(0.8)$$ $$t \cdot \ln(0.91) = \ln(0.8)$$ 5. **Isolate $t$:** $$t = \frac{\ln(0.8)}{\ln(0.91)}$$ 6. **Calculate:** $$t \approx \frac{-0.2231}{-0.0943} \approx 2.36$$ years --- 1. **Problem statement:** After authorities intervene, the fish population grows by 6% annually. We want to find the time to return to the original population of 50000. 2. **Formula for growth:** $$N(t) = N_0 \cdot (1 + r)^t$$ where $N_0$ is the population at intervention, $r=0.06$. 3. **Initial population at intervention:** From previous step, population at $t=2.36$ years is: $$N_0 = 50000 \cdot 0.91^{2.36} = 50000 \cdot 0.8 = 40000$$ 4. **Set equation to find $t$ to reach 50000 again:** $$40000 \cdot 1.06^t = 50000$$ 5. **Divide both sides:** $$1.06^t = \frac{50000}{40000} = 1.25$$ 6. **Take natural logarithm:** $$t \cdot \ln(1.06) = \ln(1.25)$$ 7. **Isolate $t$:** $$t = \frac{\ln(1.25)}{\ln(1.06)}$$ 8. **Calculate:** $$t \approx \frac{0.2231}{0.0583} \approx 3.83$$ years --- **Final answers:** - a) Fish population after $t$ years: $$N(t) = 50000 \cdot 0.91^t$$ - b) Time until authorities act: $$t \approx 2.36$$ years - c) Time to recover original population after growth starts: $$t \approx 3.83$$ years