H(t)=-0.2t^2+2t models the height in feet of a ball that is kicked into the air where t is given as time, in seconds. after how many seconds does the ball reach its maximum height? what is the maximum height of the ball? after how many seconds does the ball reach the ground?

To get the maximum height, we first determine the time at which this maximum height is attained by differentiating the given equation and equating the differential to zero.
                                 h(t) = -0.2t² + 2t
Differentiating,
                               dh(t) = -(0.2)(2)t + 2 = 0
The value of t is equal to 5. Substituting this time to the original equation,
                                   h(t) = -0.2(5²) + 2(5) = 5 ft
Thus, the maximum height is 5 ft and since it will take 5 seconds for it to reach the maximum height, the total time for it to reach the ground is 10 seconds.

Answers:              maximum height = 5 ft
                 time to reach max height = 5 seconds
            time it will reach the ground = 10 s