1. **State the problem:** We have a logistic growth model for the cat population given by $$P(t) = \frac{84}{1 + 11e^{-0.07t}}$$ where $t$ is time in weeks and $P(t)$ is the population at time $t$. We need to find: a. The carrying capacity. b. The growth rate. c. The initial population. d. The population after 5 weeks. 2. **Recall the logistic growth formula:** $$P(t) = \frac{L}{1 + Ce^{-kt}}$$ where: - $L$ is the carrying capacity (maximum population). - $C$ is a constant related to initial population. - $k$ is the growth rate. 3. **Identify parameters from the given function:** - Carrying capacity $L = 84$. - Constant $C = 11$. - Growth rate $k = 0.07$ (since the exponent is $-0.07t$). 4. **Find the initial population $P(0)$:** Substitute $t=0$: $$P(0) = \frac{84}{1 + 11e^{-0.07 \times 0}} = \frac{84}{1 + 11e^{0}} = \frac{84}{1 + 11 \times 1} = \frac{84}{12} = 7$$ 5. **Find the population after 5 weeks $P(5)$:** Substitute $t=5$: $$P(5) = \frac{84}{1 + 11e^{-0.07 \times 5}} = \frac{84}{1 + 11e^{-0.35}}$$ Calculate $e^{-0.35} \approx 0.7047$: $$P(5) = \frac{84}{1 + 11 \times 0.7047} = \frac{84}{1 + 7.7517} = \frac{84}{8.7517} \approx 9.6$$ **Final answers:** - a. Carrying capacity = 84 cats - b. Growth rate = 0.07 per week - c. Initial population = 7 cats - d. Population after 5 weeks $\approx$ 9.6 cats