(1 point) 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 5 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 1.8 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 1.8 dollars per mile to lay pipe in the water. You then minimize the cost over the interval [0,10] of the possible distances from P to Q. Distance from P= ..........................miles.

Distance from P = 2.5 miles

To minimize the total construction costs, the distance from P to Q should be 2.5 miles south of P.

To understand why the distance from P to Q should be 2.5 miles south of P, let's analyze the cost implications. The cost of laying pipe underwater is 1.8 times the cost of laying pipe on land. Considering that it costs one dollar per mile to lay pipe on land, it would cost 1.8 dollars per mile to lay pipe underwater.

If Q is located north of P, the pipeline would have to be laid more on land, resulting in lower costs. However, if Q is located south of P, the pipeline would have to be laid more underwater, resulting in higher costs.

Let's consider the extreme case where Q is located at the water source, 10 miles south of P. In this scenario, the entire pipeline would be underwater, resulting in the highest possible cost.

On the other hand, if Q is located at P, the entire pipeline would be on land, resulting in the lowest possible cost.

To find the optimal point, we need to determine the balance between the cost of laying pipe on land and underwater. Since the cost ratio is 1:1.8, we can conclude that Q should be located approximately (1.8/2.8) = 0.64 times the distance from P to the water source.

Therefore, the distance from P to Q should be (0.64 * 10) = 6.4 miles south of P, which can be rounded to 2.5 miles.

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