Let x = the distance from point B east to point Q.

The distance form the island house, H, to point Q is the hypotenuse of the right triangle HBQ.

d_{HQ} = √(x^{2} + 6^{2})

The distance from point Q to the the Water, W, is:

d_{QW} = 8-x

C(x) = 8000√(x^{2} + 36) + 5000(8-x)

To find the minimum cost, take the derivative of C(x), set that to zero and solve for x.

C'(x) = 8000(2x)/(2√(x^{2} + 36)) - 5000x

8000x/√(x^{2} + 36) - 5000 = 0

8000x = 5000√(x^{2} + 36)

8x/5 = √(x^{2} + 36)

Squaring both sides

64x^{2}/25 = x^{2}+36

39x^{2}/25 = 36

x = 6√(25/39) = 30√(1/39)≈**4.8 miles east of point B**

Note: C(4.8) ≈ $77470