Answer:
The pipeline should come ashore √2 km toward the power station from the point on shore closest to the island.
Step-by-step explanation:
Let d represent the distance from the power station to the point where the power line comes ashore. The relative cost of the line will be ...
c = 3√(4² +(8-d)²) +d
c = 3√(80 -16d +d²) +d
Differentiating c with respect to d and setting that to zero gives ...
dc/dd = 0 = (3/2)(2d-16)/√(80 -16d +d²) +1
Subtracting 1, multiplying by the denominator, and squaring both sides gives a quadratic that can be solved in the usual way. It reduces to ...
(d -8)² -2 = 0
d = 8±√2 . . . . . . . . only the root 8-√2 is useful
This tells us the pipeline should come ashore √2 km toward the power station from the point closest to the island.
