Rate of Change of Airplanes in 3D

[lisa8840]

One plane is flying due east straight and level at 30000 feet and at 420 mi/h. A second plane flies due north at 40560 feet and at 480 mi/h. The second plane crosses above the flight path of the first plane 2 minutes after the first plane passes that point. How fast were they approaching each other when they were a minimum distance apart?

 

[MarkFL]

Hello and welcome to MHB! :D

Can you show us what you have tried so we know where you are stuck and can offer better help?

 

[lisa8840]

Hello and welcome to MHB! :D

Can you show us what you have tried so we know where you are stuck and can offer better help?

I have found the time that they are at the shortest distance to be 1.13 minutes for the first plane and 0.87 minutes for the second plane. I do not know where to continue from there though. I'm also not necessarily sure how to explain that I found the time was 1.13 minutes.

 

[MarkFL]

How can you have different times for the two airplanes? You want to find the time at which the two airplane's are at the minimum distance from one another.

I would most likely orient the lower airplane in the $xy$ plane, or $z=0$ and the upper airplane in the plane $z=10560$.

I would then let the lower airplane travel along the $x$-axis ($y=0$) in the positive direction, while the upper plane travels along the line $x=0$ in the positive direction. Let the units of length be feet and of time be hours.

The point where the two flight paths cross, let's let that be the origin. So, we then define the parametric equations of motion for the two airplanes. Let time $t=0$ correspond to the time the lower airplane is at the origin:

Airplane 1:

$$x(t)=420t$$

$$y(t)=0$$

$$z(t)=0$$

Airplane 2:

$$x(t)=0$$

$$y(t)=480t-16$$

$$z(t)=10560$$

Now, you want to construct a function representing the distance between the two and minimize this function. For simplicity, you could use the square of the distance.

Actually, reading the question again, it seems to be a trick question. They are not asking how far apart they are at the minimum distance, they are asking at what rate their distance is changing at this minimum...think about it, and you will see you need no calculations at all to answer this. :D