Don't modify the equation. It is true that f(3)= 288 for f(t) = -16t^2 +144t
the maximum height of the ball will be where the derivative is zero. In other words you could find the height of the ball by converting this into vertex form, or you could find it by calculating the time needed for the ball to reach its maximum height.
0=-32t+144
t=3 till max and max is 288
224=-16t^2 +144t
0= -16t^2+144t-224
t=2 but then the ball will reach 224 ft again at t=7
the ball reaches its maximum height at t=3 seconds and a parabola is symmetric, so at t=6 seconds it will contact the ground again.