Question

Suppose that the distance an aircraft travels along a runway before takeoff is given by upper d equals left parenthesis 5 divided by 3 right parenthesis t squaredd=(5/3)t2​, where d is measured in meters from the starting point and t is measured in seconds from the time the brakes are released. the aircraft will become airborne when its speed reaches 480480 km divided by hkm/h. how long will it take to become​ airborne, and what distance will it travel in that​ time?

Answer 1

The aircraft will take 40 seconds to become airborne and it will travel 2666.666.... meters in that time.

Explanation

The distance an aircraft travels along a runway before takeoff is given by.....

$d= \frac{5}{3}t^2$ , where $d$ is in meters and $t$ is in seconds.

The aircraft will become airborne when its speed reaches 480 km/h. So first we need to convert this 480 km/h into meter/second.

$480 km/h= \frac{480*1000}{3600}meter/second = \frac{400}{3} meter/second$

Now for finding the time taken by the aircraft to become​ airborne, we need to find the value of $t$ when  $\frac{d}{dt}(d) = \frac{400}{3}$

The given equation is:  $d= \frac{5}{3}t^2$

Taking derivative both sides with respect to $t$, we will get ......

$\frac{d}{dt}(d)= \frac{5}{3}(2t)\\ \\ \frac{400}{3}=\frac{10t}{3}\\ \\ 400=10t\\ \\ t=40$

So, the aircraft will take 40 seconds to become airborne.

Now, the distance traveled in that time will be:  $d= \frac{5}{3}(40^2)= \frac{5}{3}(1600)=2666.666....$ meters.