The height function for constant acceleration is `h=-1/2 g t^2+v_0t+h_0` where `v_0` is the initial velocity and `h_0` is the initial height.  

For both children, the time to enter the water is found by setting the height to zero and solving for t.

For Hannah, we need to find the time by solving:

`-1/2(32)t^2+9.7t+24.61=0`

`-16t^2+9.7t+24.61=0`   use quadratic formula

`t={-9.7-sqrt{9.7^2+4(24.61)(16)}}/{2(-16)}`   using negative root to keep time positive

`=1.58`

For Tyler, solve the equation:

`-1/2(32) t^2+4.51t+32.81=0`   

`-16t^2+4.51t+32.81=0`   use quadratic formula

`t={-4.51-sqrt{4.51^2+4(16)(32.81)}}/{2(-16)}`

`=1.58`

Within a hundredth of a second, both children enter the water at exactly the same time of 1.58 seconds after they jump.