h=-(2t^2-7t-9)=-(2t-9)(t+1)=(9-2t)(t+1); h'=-4t+7, which is maximum at t=7/4 (h'=0). So maximum height=(11/2)(11/4)=121/8=15.125. It takes 7/4=1.75 seconds to reach maximum height. If h is the height above the water, then the stone reaches the water when h=0, 9-2t=0, t=4.5 seconds.
Alternative solution for first part (without using calculus):
h=-2(t^2-(7/2)t-9/2)=-2(t^2-(7/2)t+(7/4)^2-72/16-(7/4)^2)=-2((t-7/4)^2-121/16)=-2((t-7/4)^2-(11/4)^2) (completing the square).
So h=2((11/4)^2-(t-7/4)^2), and h is clearly maximum when (t-7/4)^2=0, its minimum value, when t=7/4=1.75 seconds and h=2*121/16=121/8=15.125.