# The minimum cost of a power line

1. Mar 17, 2009

### Emethyst

1. The problem statement, all variables and given/known data
A power line is to be constructed from a town to a mine that is located near a straight road going to the town. The point on the road closest to the mine is 7.5km from the town and 3km from the mine. The cost of building along the road is 4500$per km, and through the forest to the mine, 7500$ per km. Find the minimum cost of the power line.

2. Relevant equations
Implicit differentation

3. The attempt at a solution
No idea how to start this one. I started by drawing a right angle triangle, where the right angle is the point closest to the mine that lies on the road and the hypotenuse is the line that runs from the mine to the town. After that I an unsure how to go about this one. From the prices it seems to me that rates may be involved in this, but after trying that out by using Pythagorean's Theorem it didn't get me anywhere (though I was close to the correct answer). If anyone could help me out here it would be greatly appreciated, thanks.

2. Mar 17, 2009

### mgb_phys

Total cost = distance_road * 4500 + distance_forest * 7500
The best solution will go in a straight line to the road and then along the road to the town,
call the distance from the town that it meets the road = X

Now work out the distance from the mine to the road in terms of X.
ie the hypotonuse of a right triangle with sides 3km and (7.5km-X)

Then you have the above equation only in terms of X, and simply find the minimum.