A ball is launched from 8 feet off the ground at an initial vertical speed of 64 feet per second. It is aimed across a field at a target also 8 feet off the ground. The height of the ball at time, t, in seconds, is given by the function, h = - 16t 2 + 64t + 8.

The equation that is given, h = - 16t 2 + 64t + 8, represents the general formula of the motion which is h (t) = h0 + v0t - 16t² where h0 is the initial height (8 feet), v0 is the initial velocity (+64 ft/s) and t is time in seconds. To determine the time when the ball is at the maximum height, we need to determine the vertex of the parabola which is done as follows:

vertex = time at maximum height = - b/2a = - 64 / 2 (-16) = 2 seconds

The maximum height can be calculated by substituting the time we obtained to the function.

h = - 16t 2 + 64t + 8

h = - 16 (2) ^2 + 64 (2) + 8 = 72 feet