Problem $1)$ Vertical Curves

a$.$ Compute the minimum length of sag vertical curve that will provide $130$ m stopping sight distance for a design speed of $\backslash (80\backslash$mathrm$\{$~km$\}/\backslash$mathrm$\{$h$\}\backslash )$ at the intersection of a $\backslash (-2\mathrm{.}30\backslash \%\backslash )$ grade and $\backslash ($ a$+4\mathrm{.}80\backslash \%\backslash )$ grade.

b$.$ A crest vertical curve is being designed for a stopping sight distance of $240$ m $.$ The curve is to join a $\backslash (5\backslash \%\backslash )$ upgrade with a $\backslash (4\backslash \%\backslash )$ downgrade. Calculate the minimum length of the vertical curve assuming a driver's eye height of $1\mathrm{.}05$ m and an object height of $0\mathrm{.}38$ m $.$

c$.$ Compute the minimum length of vertical curve to provide passing sight distance of $680$ m for a design speed of $\backslash (100\backslash$mathrm$\{$~km$\}/\backslash$mathrm$\{$h$\}\backslash )$ at the intersection of a $\backslash (+2\mathrm{.}30\backslash \%\backslash )$ grade with a $\backslash (-0\mathrm{.}70\backslash \%\backslash )$ grade.