A small toy rocket follows a vertical path that can be modeled by y(t) = 5t³/6 - t⁴/12 (ft). What will be the maximum height? Also, what is the height when the acceleration is zero?

Position when v = 0 and a = 0

First, find the equation that models both the velocity and acceleration as function of time.

     v(t) = dx/dt = 5t²/2 - t³/3

     a(t) = dv/dt = 5t - t²

a) The maximum height will occur when velocity is zero (it has reached it zenith).

     v = 0 = 5t²/2 - t³/3
     t = 7.5 s

Substitute time back into the position equation gives,

     y_t=7.5 = 5 (7.5)³/6 - (7.5)⁴/12
              = 87.89 ft

b) Use the acceleration equation to find the time when the acceleration will be zero.

     a = 0 = 5t - t²
      t = 5 s (zero is not a realistic solution)

Substitute time back into the position equation gives,

     y_t=5.0 = 5 (5)³/6 - (5)⁴/12
              = 52.08 ft

So, the rocket has a negative acceleration after the rocket goes up only 52.08 ft, but it keeps climbing until it reaches its zenith of 87.89 feet.