A crest vertical curve connects a $+0\mathrm{.}97\%$ grade and a $-5\mathrm{.}15\%$ grade. The PVI is at station $45+90\mathrm{.}00$ at an elevation of $612\mathrm{.}00ft.$ The design speed is $45m\frac{i}{h}.$

Determine the minimum length of the vertical curve using the AASHTO rate of vertical curvature concept $("K"$ factors$),$ and then round the minimum length up to the next $10ft$ multiple for design of the vertical curve and associated calculations. Using this length, determine:

The station and elevation of the PVC

The station and elevation of the PVT

The station and elevation of the high point

The elevation of station $47+00\mathrm{.}00$