A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is $ a(t) = 60t $, at which time the fuel is exhausted and it becomes a freely "falling" body. Fourteen seconds later, the rocket's parachute opens, and the (downward) velocity slows linearly to $ -18\;ft/s $ in 5 seconds. The rocket then "floats" to the ground at that rate.

(a) Determine the position function $ s $ and the velocity function $ v $ (for all times $ t $). Sketch the graphs of $ s $ and $ v $.

(b) At what time does the rocket reach its maximum height, and what is that height?

(c) At what times does the rocket land?