The path of a table tennis ball after being hit and hitting the surface of the table can be modeled by the function h = -4.9t(t-0.42) where h is the height in meters above the table and t is the time in seconds. how long does it take the ball to hit the table? what is the ball's maximum height?

The height (m) at time t (s) is
h(t) = -4.9t(t - 0.42) = -4.9t² + 2.058t

When the ball hits the surface of the table, h = 0.
Therefore
-4.9t(t - 0.42) = 0
t = 0, or t = 0.42 s

The interval between the two times is 0.42 s.

At maximum height, the velocity, v, is zero.
Therefore
v(t) = h'(t) = -9.8t + 2.058 = 0
t = 0.21 s
The maximum height is
-4.9*(0.21)² + 2.058(0.21) = 0.216 m

Answers:
(a) It takes 0.42 s to hit the table.
(b) The maximum height is 0.216 m