Two cars approaching an intersection

[inner08]

Hi,

I have this problem that I can't seem to figure out. I'm taking physics in French but I will try my best to translate it.

At an instant, two cars, A and B, are 10km away from an intersection of two perpendicular roads. Car A is moving towards the East and has a speed of 30km/h whereas car B is moving towards the North at 50km/h (both of them are heading towards the intersection). Determine (a) the distance at the moment where they are the closest to one another; (b) where A and B are located when the distance is minimal?

I already have the answers but I just can't seem to figure out how to get to it.

(a) = 3.42km
(b) = B is 1.75km N and A is 2.95km W of the intersection

Any idea on how to figure this out? I was thinking to use the pythagorean theorem to find the velocity [ (30^2 + 50^2)^1/2 = 58km/h ] and then use that to find the time or something like that. I guess I don't know how to approach this type of problem. Any help would be appreciated!

 

[Plastik]

If you express the distance from Car A to the intersection as:
D_A = 10 - 30*t (where 30*t is the distance traveled by Car A as it is starting from the point of 10km away from the intersection)

And then express the distance from Car B to the intersection in a similar manner:
D_B = 10 - 50*t

Then by pythagoras expression of these two distances gives the distance between Car A and Car B. One may either use calculus to solve for t, then substitute back in for D, or use properties of parabolic functions (in terms of t) to determine the lowest value of D.

 

[inner08]

Thanks a bunch, I didn't think of doing it that way. :)