Draw a diagram. Label the points
R = resort
P = closest point
W = water source
X = place where the pipeline comes ashore
So let x = PX
Let the cost of laying pipe on land be 1. Then
the cost of laying pipe is
c = 2.3 * √(6^2+x^2) + 10-x
dc/dx = 23/√(x^2+36) - 1
c is minimum when x = 20√(3/143)
A small resort is situated on an island that lies exactly 6 miles from P, the nearest point to the island along a perfectly straight shoreline. 10 miles down the shoreline from P is the closest source of fresh water. If it costs 2.3 times as much money to lay pipe in the water as it does on land, how far down the shoreline from P should the pipe from the island reach land in order to minimize the total construction costs?
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