@tomleslie 

The most intersting part is that I have only ever gotten the same as what you suggest, 2 root and 4 root solution sets, on my local system.  Within mapleTA it seems to very-very often attain single root solution sets (which given my h-value is impossible).

When I check the variable values for these solutions and then go in and manually fix the values to the same constants then re-run, then the mapleTA  server will then find all the roots.  In other words the mapleTA server only finds a single root, on many-many occassions, however it will find both roots when the variables are fixed to the identical values that are randonly generated.  This leads me to know there is a bug with Maple running on the MapleTA server.

I corected the placement of the restart and the other suggestions regarding variable timeT (thanks for the advice Tom however the suggestions really were not very applicable to a resolution, except for possibly the ill-placed restart).  In any case I am on the hunt for a solution...

Essentially I am looking for a function with guaranteed zeros at t=0 and 90ish (f(t)=-t(t-90))... that is periodic between the zeros (f(t)=-t(t-90)(some trig function that doesn't cross the horizontal axis).  The concept is a rollercoaster type question in mapleTA (height of the coaster versus time).  One of the questions about the function is at what time(s) is the coaster 10 feet from the ground.  The major issue I am having is that for certain random generations, mapleTA only recognizes one possible input value given a specific output (between 0 and max height) despite there obviously being at least two inputs that match the output (according to the graph).

restart:
with(Optimization):
with(plots):
randomize():
a := MapleTA:-Builtin:-range(1800, 2300, 100):
b := (1/10)*MapleTA:-Builtin:-range(4, 8, 1):
timeT := MapleTA:-Builtin:-range(70, 100, 10):
f := -t*(b*t-b*timeT)^2*(cos(.15*t+4)^2-3)/a:
maxs := NLPSolve(f, t = 0 .. timeT, maximize):
maxim := maxs[1]:
graph := plot(f, t = 0 .. timeT, gridlines = true, 0 .. maxim+10, labels = [t, h(t)], labeldirections = [horizontal, vertical]);
h_given := 10;
expr := h_given-f:
answer_t := Student:-Calculus1:-Roots(expr, t = 0 .. timeT);
evalf(answer_t, 2);