We can find the appropriate height of the fence at a distance d from home plate by defining

Input := 

ht[dist_] := vert[Solve[horiz[t]==dist,t][[1,1,2]]]

Input := 

ht[325]

Output =

                       5 Pi
-99.1739 + 358.598 Sin[----]
                        36

Input := 

a=Plot[ht[dist],{dist,325,467},
	PlotStyle -> AbsoluteThickness[2]];