# Science:Math Exam Resources/Courses/MATH102/December 2016/Question B 07 (c)/Solution 1

From the Hint, we find ${\displaystyle F\geq 250}$ such that ${\displaystyle N_{0}=500\ln \left({\frac {F}{250}}\right)=1}$.
${\displaystyle 500\ln \left({\frac {F}{250}}\right)=1\iff \ln \left({\frac {F}{250}}\right)={\frac {1}{500}}\iff {\frac {F}{250}}=e^{\frac {1}{500}}\iff F=250e^{\frac {1}{500}}}$
Therefore, the optimal fine is ${\displaystyle \color {blue}F=250e^{\frac {1}{500}}(dollars)}$