Set 15: Exponential Functions (Advanced)

Choice A is correct. Choice A is the correct answer. Solve for $t$ when $R(t) = 500$. 1. Equation: $500= \frac{1000}{1 + 999 e^{-2 t}}$. 2. Multiply: $500(1 + 999 e^{-2 t}) = 1000$. 3. Simplify: $1+ 999 e^{-2 t} = 2$. 4. Isolate: $999e^{-2 t} = 1$, so $e^{-2 t} = \frac{1}{999}$. 5. Natural log: $-2 t = \ln(\frac{1}{999}) = -\ln(999) \approx -6.907$. 6. Solve: $t = \frac{6.907}{2} \approx 3.45 \approx 3.5$ days. Strategic Tip: Half of carrying capacity in logistic models 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.