Answer:
It will take 90s for the aircraft to become airbone.
In that time, the aircraft will have traveled 13500m = 13.5km.
Step-by-step explanation:
The velocity function is the derivative of the distance function.
In this problem, we have the following distance function:
![D(t) = \frac{5t^{2}}{3}](https://tex.z-dn.net/?f=D%28t%29%20%3D%20%5Cfrac%7B5t%5E%7B2%7D%7D%7B3%7D)
So the speed function is the following:
![S(t) = \frac{10t}{3}](https://tex.z-dn.net/?f=S%28t%29%20%3D%20%5Cfrac%7B10t%7D%7B3%7D)
The aircraft will become airborne when its speed reaches 300km/h. How long will it take to become airborne?
This is going to happen when S(t) = 300. So:
![S(t) = \frac{10t}{3}](https://tex.z-dn.net/?f=S%28t%29%20%3D%20%5Cfrac%7B10t%7D%7B3%7D)
![\frac{10t}{3} = 300](https://tex.z-dn.net/?f=%5Cfrac%7B10t%7D%7B3%7D%20%3D%20300)
![t = 90](https://tex.z-dn.net/?f=t%20%3D%2090)
It will take 90s for the aircraft to become airbone.
What distance will it travel in that time?
This is D(t) when t = 90. So:
![D(t) = \frac{5t^{2}}{3}](https://tex.z-dn.net/?f=D%28t%29%20%3D%20%5Cfrac%7B5t%5E%7B2%7D%7D%7B3%7D)
![D(90) = \frac{5*90^{2}}{3} = 13500](https://tex.z-dn.net/?f=D%2890%29%20%3D%20%5Cfrac%7B5%2A90%5E%7B2%7D%7D%7B3%7D%20%3D%2013500)
In that time, the aircraft will have traveled 13500m = 13.5km.