A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is $a(t) = 60t \,\,ft/s^2,$ at which time the fuel is exhausted and it becomes a freely "falling" body. Seventeen seconds after the model rocket is fired, the rocket's parachute opens, and the (downward) velocity slows linearly to -18 ft/s in five seconds. The rocket then floats to the ground at that rate.

(a) Find the velocity function $v(t)$ (in ft/s) for each of the following time intervals:

$\text{If } 0 \le t \le 3, v(t) =$ ft/s

$\text{If } 3 < t \le 17, v(t) =$ ft/s

$\text{If } 17 < t \le 22, v(t) =$ ft/s

$\text{If } t > 22, v(t) =$ ft/s

(b) Find the position function $s(t)$ (in ft) for each of the following time intervals:

$\text{If } 0 \le t \le 3, s(t) =$ ft

$\text{If } 3 < t \le 17, s(t) =$ ft

$\text{If } 17 < t \le 22, s(t) =$ ft

$\text{If } t > 22, s(t) =$ ft

(c) At what time (in seconds) does the rocket reach its maximum height, and what is that height (in feet)?

Time at which maximum height is reached = seconds

Maximum height = ft

(d) At what time (in seconds) does the rocket land?

Landing time = seconds

You can earn partial credit on this problem.