You can put this solution on YOUR website!
So, the maximum height is when h is at it's maximum in h=68t-8t^2. There's a lot of ways to find this out.
One could, for example, take the first derivative, which would be 68-16t which is 0 when t=17/4, so the maximum is 68(17/4)-8(17/4)=255.
Another way would be to take advantage of the symmetry of quadratics, and say that the two zeros of 68t-8t^2 are 0 and 17/2, so the maximum occurs halfway in between them, or at 17/4.
Another way is the generalization you may have learned that the vertex is at the x value -b/2a, which is this case is -68/-16=17/4. Lots of ways.