Question 390217
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Hi
h = 22t - 5t^2   Path of the stone is a parabola:
I. with y-intercept at Pt(0,0)
II.  Opening downward (Note: negative coefficient for the x^2 term)
Note: the vertex form of a parabola, {{{y=a(x-h)^2 +k}}} where(h,k) is the vertex
h = 22t - 5t^2 completing square to find the vertex
h = -5(t^2 - 22/5)
h = -5[(t - 11/5)-121/25]
h = -5( t - 11/5)^2 + 121/5  
Vertex (maximum point) is ( 11/5,121/5)or(2.2,24.2) 
Line of symmetry for this parbola being the perpendicular line x = 2.2
a) time to reach the maximum height is 2.2 sec
b) time to reach the surface of the lake: 0 = t(-5t + 22)  t = 22/5 = 4.4 sec
Sketching the graph to see the path the stone thrown takes:
I. Plot the Points: y-intercept Pt(0,0), vertex (2.2,24.2)
II. draw the vertical axis of symmetry  x = 2.2 and then Plot Pt(4.4,0)
III. Sketch the parabola opening downward using the axis of symmetry as a reference
{{{drawing(300,300, -30, 30, -30, 30,blue(line(2.2,-30,2.2,30.2)), grid(1),
circle(2.2, 24.2,0.9),
circle(4.4, 0,0.9),
circle(0, 0,0.9),

graph( 300, 300,   -30, 30, -30, 30,-5x^2 + 22x))}}}