h=192t-32t^2

192 ft/s the initial velocity

h = the height

t = the time in seconds

a. Find the maximum height attainted by the object

We need to find the maximum of the parabolic function:

h=-32t^2 + 192t

The quotient in front of t^2 is -32<0 follows the function has maximum equal to:

h max = c - b^2/(4a), where a = -32, b = 192, c = 0

h max = 0 - 192^2/(4*(-32))

h max = 288 ft

b. Find the number of seconds it takes the object to hit the ground.

The object hits the ground when h = 0 for some t > 0 that is:

-32t^2 + 192t = 0

........

........

t = 6 s

It takes 6 seconds for the object to hit the ground.