How Do You Determine the Optimal Station Location Between Two Towns?

[asemh]

Hi, I am having a hard time with this Optimization question as i do not know where to begin, I drew a diagram but what formulas, function etc do I use to start the question? And How do i do it?

Two towns A and B are 7km and 5km, respectively, from a railroad line. The points C and D nearest to A and B on the line are 8 km apart. Where should the station be located to minimize the length of a new road from A to S to B?

Here is the Diagram i drew:

http://img109.imageshack.us/img109/7071/diagramv.png

How would I start going about doing this problem?

 

[Office_Shredder]

Can you write the length of the road as a function of where S is? (in particular, think of S as being the distance from point C)

 

[asemh]

See I can't tell if Point CS = 4km and point SD = 4km
I am not sure if they are both equivalent or not?
Im confused at this point.

I know the diagram is right because my teacher was helping me on that part.

 

[hotvette]

I think you are trying to jump ahead. S (i.e. distance from C to the station as Office Shredder suggested) is the unknown. By the diagram you drew, the optimal value of S is somewhere between 0 and 8km. If you write an equation L = f(S) where L is the length of the road you are trying to minimize, then it's a matter of setting the derivative of L with repect to S equal to zero and solving for S.