Question

applications

1. a toy rocket is fired into the air. its height, $h$, is a function of the time, $t$, since the rocket was launched. this function is shown in the graph below.

(a) what is the maximum height the rocket reaches? use appropriate units in your answer.
(b) for how many seconds is the rocket in flight?
(c) how would you interpret the statement that $h(3)=142$?
(d) give an interval over which this function is only increasing. what is happening to the rocket during this interval?
(e) give a time when the height of the rocket is the same as it was two seconds after its launch.
height above the ground (meters)
time since launched (seconds)

Explanation:

Step1: Identify max height from graph

The peak of the parabola is at $h=150$ meters.

Step2: Find total flight time

The rocket launches at $t=0$ and lands at $t=12$ seconds, so flight time is $12-0=12$ seconds.

Step3: Interpret $h(3)=142$

$h(t)$ is height at time $t$, so this means at 3 seconds, height is 142 meters.

Step4: Find increasing interval

The function rises from launch to the peak, so $t$ from 0 to 6 seconds. During this time, the rocket is ascending (gaining height).

Step5: Find matching height time

At $t=2$, height matches another time. From symmetry of the parabola, the peak is at $t=6$, so $6+(6-2)=10$ seconds.

Answer:

(a) 150 meters
(b) 12 seconds
(c) 3 seconds after launch, the rocket is 142 meters above the ground.
(d) Interval: $0 < t < 6$ seconds; the rocket is ascending (gaining height).
(e) 10 seconds