The problem is says h is quadratic in t
so h = At^2 + Bt + C
First thing is to find A, B and C
What do you know?
In the beginning h = 10m and t = 0s
when h = 11.225m, t = 0.5s
Now you have 3 unknowns so there needs to be one more piece of information. I think that this requires knowing Earth's gravity is about 10 m/s^2 and the coefficient A is half of that so A = 5 m/s^2 (there should be an example in your text book).
How much time was Alex above water?
When h = 0 what is t?
Remember t is always positive.