There's a lot of extra information here. All you are really concerned with is solving the polynomial for roots. You want t when h = 14. So let's do it:
[tex]14 = 6+23t-16t^2[/tex]
[tex]8=23t-16t^2[/tex]
Rearranging to make it simpler to put into a quadratic form:
[tex]0 = -16t^2+23t-8[/tex]
Now, let's plug it into the quadratic equation:
[tex] \frac{-23+ \sqrt{(23)^2-4(-16)(-8)} }{2(-16)} [/tex]
and
[tex]\frac{-23- \sqrt{(23)^2-4(-16)(-8)} }{2(-16)}[/tex]
Solving the quadratic gives us:
0.5899 and 0.8476
So, your answer is: h = 14 when t = 0.5899 or t = 0.8476
