we have the equation
H = -16t^2 + 36t + 56
This equation represents a vertical parabola open downward, which means, the vertex is a maximum
The time t when the ball reaches its maximum value corresponds to the x-coordinate of the vertex
so
Convert the given equation into vertex form
H=a(t-h)^2+k
where
(h,k) is the vertex
step 1
Complete the square
H = -16t^2 + 36t + 56
Factor -16
H=-16(t^2-36/16t)+56
H=-16(t^2-36/16t+81/64)+56+81/4
Rewrite as perfect squares
H=-16(t-9/8)^2+76.25
the vertex is (9/8,76.25)
therefore
the time is 9/8 sec or 1.125 seconds when the ball reaches its maximum