Help with math (Quadratic Functions)

Question

An underground silo contains a rocket which releases a weather balloon after the rocket reaches its maximum height. The rocket then plummets back to Earth. The height of the rocket is modeled by
h(t) = -4.9t^2 + 58.8t - 50 Where h(t) is the height of the rocket in meters and t is the time in seconds.

1. How far below ground is the silo?

2. At what time is the rocket first above ground, to the nearest tenth of a second?

3. At what time does the rocket land back on Earth, to the nearest tenth of a second?

4. Relate these two times (When the rocket is first above ground and when it lands on the ground) to intervals in which the function is positive or negative.

Please answer whichever you can. Any help is very much appreciated. :)

Answer 1

(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).