From the Hint, we find F ≥ 250 {\displaystyle F\geq 250} such that N 0 = 500 ln ( F 250 ) = 1 {\displaystyle N_{0}=500\ln \left({\frac {F}{250}}\right)=1} .
By some simple computation, we get
500 ln ( F 250 ) = 1 ⟺ ln ( F 250 ) = 1 500 ⟺ F 250 = e 1 500 ⟺ F = 250 e 1 500 {\displaystyle 500\ln \left({\frac {F}{250}}\right)=1\iff \ln \left({\frac {F}{250}}\right)={\frac {1}{500}}\iff {\frac {F}{250}}=e^{\frac {1}{500}}\iff F=250e^{\frac {1}{500}}}
Therefore, the optimal fine is F = 250 e 1 500 ( d o l l a r s ) {\displaystyle \color {blue}F=250e^{\frac {1}{500}}(dollars)}