1. **State the problem:** We have a logistic growth model for the cat population given by $$P(t) = \frac{84}{1 + 11 e^{-0.07t}}$$ where $t$ is in weeks. 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 model formula:** $$P(t) = \frac{K}{1 + Ae^{-rt}}$$ where: - $K$ is the carrying capacity (maximum population) - $r$ is the growth rate - $A$ is a constant related to initial conditions 3. **Identify parameters from the given model:** Comparing, $$K = 84$$ $$A = 11$$ $$r = 0.07$$ 4. **a) Carrying capacity:** The carrying capacity $K$ is the maximum population the environment can support. From the formula, $K = 84$. 5. **b) Growth rate:** The growth rate $r$ is the coefficient in the exponent. From the formula, $r = 0.07$ per week. 6. **c) Initial population $P(0)$:** Calculate $P(0)$ by substituting $t=0$: $$P(0) = \frac{84}{1 + 11 e^{-0.07 \times 0}} = \frac{84}{1 + 11 \times 1} = \frac{84}{12} = 7$$ 7. **d) Population after 5 weeks $P(5)$:** Calculate $P(5)$: $$P(5) = \frac{84}{1 + 11 e^{-0.07 \times 5}} = \frac{84}{1 + 11 e^{-0.35}}$$ 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