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