Minimum Cost for pipe from offshore well to refinery

[cummings]

The problem is: An offshore oil well is 2 kilometers off the coast. The refinery is 4 kilometers down the coast. Laying pipe in the ocean is twice as expensive as on land. What path should the pipe follow in order to minimize the cost?

I know I could take the path as a rectangular but that would not the be the cheapest, neither would a straight diagonal line between the two points because all the piping would go through the ocean. I assume that I am looking for a diagonal to cut across the corner and hit the coast and a straight that will take me to the refinery. However to find the shortest distance for the diagonal is what I am struggle with. I know that there are two different rate of changes depending on where I am at. So any help would be wonderful.

 

[skeeter]

Re: Minimum Cost

make a sketch ...

let x = distance (in km) downshore where the underwater pipe hits land ... (4-x) is the remaining downshore distance in km to the refinery.

length of underwater pipe in km = \(\displaystyle \sqrt{x^2 + 4}\)

length of land pipe in km = \(\displaystyle 4 - x\)

now ... come up with a Cost function, find \(\displaystyle \frac{dC}{dx}\) and minimize.