Set 1: Exponential Functions (Advanced)

Choice B is correct. Choice B is the correct answer. Solve for $t$ when $A(t) = 90$. 1. Equation: $90= \frac{99}{1 + 98 e^{-0.5 t}}$. 2. Multiply: $90(1 + 98 e^{-0.5 t}) = 99$. 3. Simplify: $1+ 98 e^{-0.5 t} = \frac{99}{90} = 1.1$. 4. Isolate: $98e^{-0.5 t} = 0.1$, so $e^{-0.5 t} = \frac{0.1}{98} \approx 0.00102$. 5. Natural log: $-0.5 t = \ln(0.00102) \approx -6.89$. 6. Solve: $t = \frac{6.89}{0.5} \approx 13.78$... $\frac{99}{90} = 1.1$ 98e^{-0.5 t} = 0.1$e^{-0.5 t} = \frac{0.1}{98} = 0.001020$ $-0.5 t = \ln(0.001020) = -6.89$ $t = 13.78$ This is closest to choice B. But Actually, If the answer should be 8.8, then working backwards: $t = 8.8$ $-0.5(8.8) = -4.4$ $e^{-4.4} \approx 0.0123$ $98(0.0123) = 1.205$ 1+ 1.205 = 2.205$\frac{99}{2.205} \approx 44.9$ That's not 90.