2. a rocket is launched from a cliff. the relationship between the height of the rocket, ℎ, in feet, and the time since its launch, , in seconds, can be represented by the following function: ℎ() = −162 + 80 + 384. a. sketch the graph of the motion of the rocket. b. when does the rocket hit the ground? c. when does the rocket reach its maximum height? d. what is the maximum height the rocket reaches? e. at what height was the rocket launched?

The model that represents the relationship between the height of the rocket and time is given by:
h(t)=-16t^2+80t+384
b]When does the rocket hit the ground?
when the rocket hits the ground h(t)=0
thus:
0=-16t^2+80t+384
solving the above quadratic we get:
t=-3 or t=8
since there is not negative time, we conclude that the rocket will hit the ground after 8 seconds.

c] When does the rocket reach its maximum height?
when the rocket riches the maximum time, dh/dt=0
thus differentiating our function and equating it to zero then solve for t we get:
-32t+80=0
t=80/32
t=2.5
thus the rocket reached the maximum height after 2.5 sec

d] The maximum height will be as follows:
time taken to reach maximum height is t=2.5
thus
h(2.5)=-16(2.5)^2+80(2.5)+384
h(2.5)=484 ft
Maximum height is at 484 ft.

e] At what height was the rocket launched?
at the time the rocket was launched, t=0
thus
h(0)=-16(0)^2+80(0)+384
h(0)=384 ft
The rocket was launched at the height of 384 ft

a] The sketch of the function will be: