Problem $1$: $(15$ points$).$ A $+0\mathrm{.}4\%$ tangent intersects at $-0\mathrm{.}6\%$ tangent at station $80+00$ and elevation $500ft.$ On an airport runway the minimum length vertical curve required is $1000ft.$ a$)$ Find the coordinates of the highest point on that minimum length curve. b$)$ Find the equation of the longest curve that will allow this runway to intersect another runway at station $78$ and elevation $498ft.$