Solution
= Solution
For $x=\alpha T\ll1$, $e^{-x}(1-e^{-x})=x-\tfrac32x^2+O(x^3)$. Hence $\tfrac32\beta_{\max}\alpha T^2\sim\log100$ and
$$
\boxed{T\sim\sqrt{\frac{2\log100}{3\beta_{\max}\alpha}}.}
$$
= Solution
For $x=\alpha T\ll1$, $e^{-x}(1-e^{-x})=x-\tfrac32x^2+O(x^3)$. Hence $\tfrac32\beta_{\max}\alpha T^2\sim\log100$ and
$$
\boxed{T\sim\sqrt{\frac{2\log100}{3\beta_{\max}\alpha}}.}
$$