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
