Two towns A and B are 5km and 7km, 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 S to B?
So I set up the question in order to minimize the length of AS^2+SB^2 (since we are not suppose to use trig). I squared the sides in order to make the problem easier to work with. I made x be the length from C to S, and 6-x the length from S to D.
(5^2 + x^2) + (7^2 + (6-x)^2) = 110 - 12x + 2x^2
differentiating, I get,
When I solve for x, I get 3. However, the answer in the back of the book is 21/6=3.5 from point D.
What am I doing wrong?