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
