Question

The cost of laying power line underwater is 3 times that of underground. An island is 4km from the shore and a power station is at a distance of 8km from the point on the shore which is closest to the island. How should the power line be laid so that the cost is a minimum?

Answer 1

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.