Question

2. If a toy rocket is launched vertically upward from ground level with an initial velocity of 128 feet

per second, then its height h after t seconds is given by the equations h(t) = -16° +128t (if air

resistance is neglected).

a. How long will it take for the rocket to return to the ground?

b. After how many seconds will the rocket be 112 feet above the ground?

C. How long will it take the rocket to hit its maximum height?

d. What is the maximum height?

Answer 1

a. 8 seconds.

b. 1 second and 7 seconds.

c. 4 seconds.

d. 256 feet.

Explanation:

The given function is

`h(t) = -16t^(2)+128t`

The image attached shows the trajectory of the rocket in a height-time graph. Notice that the maxium point reached is (4, 256), which means after 4 seconds, the rocket has a maximum height of 256 feet. (c) and (d)

Additionally, a parabola is symmetrical, which means it takes the same time going up or down, therefore the rocket reaches the ground after 8 seconds. (a)

Now, when the rocket is 112 feet above the ground, we have

`112 = -16t^(2)+128t`

Where we need to solve for
`t`

`0 = -16t^(2)+128t-112`

Using a calculator, we have the solutions
`x_(1)= 1` and
`x_(2)=7`, which means after 1 second and 7 second, the rocket is 112 feet above the ground. (b)