PROBLEM $3$

A vertical curve is to be designed to join a $+3\mathrm{.}5\%$ grade to a $-2\mathrm{.}5\%$ grade. The elevation of PVI is $312$ m $.$ If the design speed is $100$ kph $,$ determine:

$($a$)$ The design length of the curve.

$($b$)$ The elevation of the beginning of the curve.

$($c$)$ The elevation of the end of the curve.

$($d$)$ The location and elevation of the maximum point on the curve.