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

The rocket can safely explode in 10 ≤ t ≤ 30 seconds (see the last line of explanation).

Explanation:

The height of the rocket is shown by
`s = 200t - 5t^(2)`, where t is the time in seconds.

`(ds)/(dt) = 200 - 10t`. The height will be maximum when,
`(ds)/(dt) = 0` that is
`t = (200)/(10) = 20`.

At t = 20, s = 4000 - 2000 = 2000.

The rocket can go 2000 meter maximum.

In order to explode safely, the rocket must be in a height equal or above 1500 meter.

When s = 1500,
`1500 = 200t - 5t^(2) \\t^(2) - 40t + 300 = 0\\t^(2) - 30t - 10t + 300 = 0\\t = 30,10.`

Hence, at
`10 \leq t \leq 30`, the height
`s \geq 1500`.