Question

A small resort is situated on an island that lies exactly 3 miles from pp, the nearest point to the island along a perfectly straight shoreline. 10 miles down the shoreline from pp 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 pp should the pipe from the island reach land in order to minimize the total construction costs?

Answer 1

Suppose the pipe goes underwater directly from the resort to point P, which is x miles away from the point on the shoreline that's closest to the resort. By Pythagoras, the distance from the resort to P is: sqrt(x^2 + 3^2) = sqrt(x^2 + 9) Suppose that it costs 1 unit of money per mile to lay pipe on land, so therefore it costs 1.5 units of money to lay pipe underwater. So the cost of the pipe above is: 1.5*sqrt(x^2 + 9) So the distance from P to the fresh water source is 10 - x, so the cost of laying that pipe is 10 - x. Total cost of pipe: C = 1.5*sqrt(x^2 + 9) + 10 - x dC/dx = ((1.5x) / sqrt(x^2 + 9)) - 1 To minimize C, set the derivative to zero: ((1.5x) / sqrt(x^2 + 9)) - 1 = 0 (1.5x) / sqrt(x^2 + 9) = 1 1.5x = sqrt(x^2 + 9) (1.5x)^2 = x^2 + 9 2.25x^2 = x^2 + 9 2.25x^2 - x^2 = 9 1.25x^2 = 9 x^2 = 9 / 1.25 x = sqrt(9/1.25) x = 2.683