(1) When t=0 (before takeoff) h(0)=-50, so the silo is 50m below ground.
(2) When h(t)=0, the rocket clears the ground, so:
-4.9t2+58.8t-50=0, 4.9t2-58.8t+50=0, using the quadratic formula:
t=(58.8±√(58.82-980))/9.8=(58.8±√2477.44)/9.8=(58.8±49.774)/9.8.
We need the smaller value because the other value is when the rocket falls back to the ground.
t=0.9 seconds approx.
(3) The rocket returns to earth after (58.8+49.774)/9.8=11.1 seconds approx.
(4) The rocket reaches its maximum height midway in the interval 0.9 to 11.1 (58.8/9.8=6 seconds). The function value (height) increases up to when t=6 seconds, then it decreases. The function is negative up to t=0.9 seconds and after t=11.1 seconds. This is because outside this time the rocket would be below ground (theoretically, it returns to the silo).