Application of Differentiation Problem

[bjgawp]

Here's a problem that I just found in my book and to my dismay, I couldn't figure out how differentiation can be used to solve a particular problem (seeing how we've just finished this unit at school). So here's the problem:

Felicity and Jane start alking at the same time towards an intersection of two roads that meet at right angles.

http://img528.imageshack.us/img528/66/untitledfh4.png

Felicity starts at 9km from the intersection while Jane starts at 13km from the intersection. Their speeds are 4 km/h and 3 km/h respectively. What is the closest that Felicity and Jane will get?

I cannot figure out how to relate the two into one equation. Obviously, we need an equation for the distance between them and find the minimum for it (i.e. f'(x) = 0). Anyway, I thought I got differentiation down pat but ... guess not

 

[arildno]

Well, denote Felicity's position VECTOR as a function of time:
$$\vec{r}_{F}(t)=(x_{F}(t),y_{F}(t))$$
and similarly, for Jane:
$$\vec{r}_{J}(t)=(x_{J}(t),y_{J}(t))$$

now, choose an intelligent origin, specify the component functions, and find an expression for the distance between them, as a function of time.

 

[bjgawp]

I haven't learned anything about vectors but I think I've got it. x = Felicity's distance to the intersection while y = Jane's distance to the intersection.

http://img379.imageshack.us/img379/5775/problemns7.jpg

So the closest they can get is 5 km Phew, haven't lost my touch yet. Thanks anyway arildno!

Hmm I wonder why the images won't actually show up ...