Question

Consider an organism growing according to S(t)=S(0)e^at. Suppose a=0.001/s, and S(0)= 1 mm. At time 1000 s, S(t)=2.71828 mm. How close must t be to 1000s to guarantee a size within 0.1 mm of 2.71828mm?

Answer 1

\[S(t) = S_0e^{at}\]You're looking to find the values of \(t\) such that \[S(t) = 2.71828\pm0.1\]
\[S_0e^{at} = S(t)\]\[e^{at} = \frac{S(t)}{S_0}\]Take natural log of both sides\[at \ln e = \ln \frac{S(t)}{S_0}\]\(\ln e = 1\), divide both sides by a and evaluate\[t = \frac{1}{a}\ln\frac{S(t)}{S_0} =\frac{1}{0.001s^{-1}}\ln\frac{2.71828mm-0.1mm}{1mm} =\ln(2.61828)*1000s = \]For the second value, use 2.81828 instead of 2.61828.