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 6 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 1.6 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?
Hint: You can do this problem by assuming that it costs one dollar per mile to lay pipe on land, and 1.6 dollars per mile to lay pipe in the water. You then need to minimize the cost over the interval [0,10] of the possible distances from P to Q.
The money derivatives over land is $1 per mile, and the money derivative over water is $1.6 per mile. x has to be between 0 and 10, the maximum possible distance underwater is sqrt(136) miles. However, that will have a rather high cost. I know that the distance will be closer to 0 than to 10. The equation will most likely be a positive quadratic equation. How will I find this equation, the distance, and the cost at this location.