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An online learning platform observes that course completion rates follow the logistic model \( C(t) = \frac{1000}{1 + 9e^{-0.2t}} \), where \( C(t) \) is the number of completions after \( t \) weeks. After how many weeks does the number of completions first exceed 900?
Solve \( \frac{1000}{1 + 9e^{-0.2t}} > 900 \)
Divide both sides:
\frac{1000}{900} > 1 + 9e^{-0.2t} \Rightarrow \frac{10}{9} > 1 + 9e^{-0.2t}
\frac{10}{9} - 1 > 9e^{-0.2t} \Rightarrow \frac{1}{9} > 9e^{-0.2t}
\frac{1}{81} > e^{-0.2t}
Take natural log:
\ln\left(\frac{1}{81}\right) > -0.2t \Rightarrow -\ln(81) > -0.2t
\ln(81) < 0.2t
\( \ln(81) = \ln(3^4) = 4\ln(3) \approx 4 \times 1.0986 = 4.3944 \)