Where do I even start?
If you don't know calculus you can do this with algebra. They show how here
This question is expected to be solved using calculus. I think it might be a maximum problem:
A stone is thrown vertically upward and after t seconds its height y meters above its starting point is given by y=100t-5t^2. Calculate the greatest height reached by the stone, and the value of t when this happens.
Yeh. Any time they want the max or the min it's a max-min problem. Do the vertex just like you always do for parabolas.