A small resort is situated on an island that lies exactly 5 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 1.5 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?

Answer 1

If the distance is x from P, then the cost is

c(x) = 1.5√(x^2+25) + (10-x)
dc/dx = 1.5x/√(x^2+25) - 1
= [1.5x - √(x^2+25)]/√(x^2+25)

dc/dx=0 when the numerator is zero, or

1.5x = √(x^2+25)
2.25x^2 = x^2+25
1.25x^2 = 25
x^2 = 20
x = 2√5