Question

During a fireworks show, a rocket is fired vertically upward with an initial velocity of 200 meters per second. The height in meters, s, of the rocket in t seconds may be approximated by s = 200t – 5t2. The rocket must be at least 1,500 meters in the air to safely explode. In which time interval may it safely be exploded
30 ≤ t ≤ 40 seconds
0 ≤ t ≤ 10 seconds
10 ≤ t ≤ 40 seconds
10 ≤ t ≤ 30 seconds

Answer 1

Answer:

In the time interval of 10 to 30 seconds the rocket may safely be exploded.

Step-by-step explanation:

The height of the rocket is shown by $s = 200t - 5t^{2}$.

$\frac{ds}{dt} = 200 - 10t$ = 0 gives t = 20 seconds.

The highest height that the rocket can go is at t = 20.

The rocket must be at least 1500 meters in the air to safely explode.

Hence, if s = 1500, then,

$1500 = 200t - 5t^{2} \\t^{2} - 40t + 300 = 0\\t^{2} - 30t - 10t + 300 = 0\\(t - 30)(t - 10) = 0\\t = 30, 10$.

After 10 seconds and 30 seconds of firing, the rocket will be in a height of 1500 meter.

Hence, the time interval of safely explode is 10 ≤ t ≤ 30 seconds.