s = 80t - 16t^2
One way to do it:
The ball starts at a height of 0 and ends at a height of 0. Find when that happens.
0 = 80t - 16t^2
0 = t(80 - 16t)
80 - 16t = 0
80 = 16t
t = 5
The ball will return to the ground at t = 5 seconds. The maximum height will happen halfway between the start at t = 0 and the end at t = 5.
Answer: 2.5 seconds.
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Another way:
s = 80t - 16t^2 is an upside-down parabola. The maximum height happens at the top of the parabola, where the parabola is flat (slope = 0, first derivative = 0).
s ' = 80 - 32t
0 = 80 - 32t
32t = 80
t = 2.5
Answer: 2.5 seconds.