- #1

shanshan

- 24

- 0

## Homework Statement

Two towns, A and B are 5 km and 7 km respectively, from a railroad line. The points C and D nearest to A and B on the line are 6 km apart. Where should a station be located to minimize the length of a new road from A to B?

## Homework Equations

a^2+b^2 = c^2, a=0.5bh(?)

## The Attempt at a Solution

I thought for this question that I could use pythagorus' theorum to find the value of c. If I considered CS as x, and SD as x-6, then I could find c in terms of x. So I've got that

c(1)=sqrt(x^2-12x+85), and

c(2)=sqrt(25+x^2)

and then I thought maybe I could add these two c values together and find the derivative of their sum, and make that equal to 0 (because I am looking for a minimum value).

Of course, this method of solving, if it is even right, is a lot more work than most of the other questions around it, which makes me wonder if there a simpler way of doing the question, or if my method even makes sense..

Does this solution make any sense? Can I use the derivative of a line's length to find it's minimum? Or are there restrictions on what type of function I can differentiate to find a minimum?