Tough (for me) optimizations problem
Question: A small resort is situated on an island off a part of the coast of Mexico that has a perfectly straight north-south shoreline. The point P on the shoreline that is closest to the island is exactly 3 miles from the island. Ten miles south of P is the closest source of fresh water to the island. A pipeline is to be built from the island to the source of fresh water by laying pipe underwater in a straight line from the island to a point Q on the shoreline between P and the water source, and then laying pipe on land along the shoreline from Q to the source. It costs 1.7 times as much money to lay pipe in the water as it does on land. How far south of P should Q be located in order to minimize the total construction costs?
My (little) work is attached.
(w = water, L = land, d = distance)
Any help would be greatly appreciated!
Thanks in advance!