Set 15: Exponential Functions (Advanced)

Choice A is correct. Choice A is the correct answer. Solve for $t$ when $I(t) = 5000$. 1. Equation: $5000= \frac{10000}{1 + 9999 e^{-1.5 t}}$. 2. Multiply: $5000(1 + 9999 e^{-1.5 t}) = 10000$. 3. Simplify: $1+ 9999 e^{-1.5 t} = 2$. 4. Isolate: $9999e^{-1.5 t} = 1$, so $e^{-1.5 t} = \frac{1}{9999}$. 5. Natural log: $-1.5 t = \ln(\frac{1}{9999}) = -\ln(9999) \approx -9.21$. 6. Solve: $t = \frac{9.21}{1.5} \approx 6.14 \approx 6.1$ days. Strategic Tip: In logistic models, half of carrying capacity occurs when $Ae^{-rt} = 1$. Choice B is incorrect because this is too long. Choice C is incorrect because this is too short. Choice D is incorrect because this is way too long.