A small resort is situated on an island that lies exactly 4 miles from , the nearest point to the island along a perfectly straight shoreline. 10 miles down the shoreline from is the closest source of fresh water. If it costs 1.5 times as much money to lay pipe in the water as it does on land, how far down the shoreline from should the pipe from the island reach land in order to minimize the total construction costs?

Question

A small resort is situated on an island that lies exactly 4 miles from , the nearest point to the island along a perfectly straight shoreline. 10 miles down the shoreline from  is the closest source of fresh water. If it costs 1.5 times as much money to lay pipe in the water as it does on land, how far down the shoreline from  should the pipe from the island reach land in order to minimize the total construction costs?

Steve · Answer

Draw a diagram, where
R is the resort
P is the named point on shore
W is the water well
L is the point on land where the pipe comes ashore

RPL is a right triangle, where
RP=4
PL = x
PW=10, so LW=10-x

without loss of generality, we may assume the land cost is 1, so the total cost is

c = 1.5√(16+x^2) + (10-x)
dc/dx = 1.5x/√(16+x^2) - 1
dc/dx = 0 when x = 8/√5 = 3.58