Question

A power line is to be constructed from a power station at point A to an island at point C, which is 1 mi directly out in the water from a point B on the shore. Point B is 4 mi down the shore from the power station at A. It costs $5000 per mille to lay the power line underwater and $3000 per mile to lay the underground. At what pointS down shores from A between the cities?

Answer 1

point S is 3 1/4 miles from point A

Explanation:

We assume you want to find point S that minimizes the power line cost.

Let k = 5000/3000 = 5/3 represent the ratio of construction cost underwater to construction cost downshore. Then the total cost will be proportional to ...

cost ~ k(CS) + SA

The distance BS can be found from the Pythagorean theorem. In miles, the relation is ...

BS² +BC² = CS²

BS² +1 = CS²

So our cost is ...

cost ~ k(CS) +BA -BS = k√(BS² +1) +BA -BS

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To minimize the cost, we want to find BS that makes the derivative d(cost)/dBS = 0.

d(cost)/dBS = kBS/√(BS² +1) -1 = 0

kBS = √(BS² +1) . . . . . . add 1, multiply by the root

k²BS² = BS² +1 . . . . . . . square

(k² -1)BS² = 1 . . . . . . . . . subtract BS² and factor

BS = 1/√(k² -1) . . . . . . . . divide by the coefficient of BS² and square root

Please note that this can be considered to be a generic solution to problems of this sort. BS is measured as a multiplier of distance BC, which in this case is 1 mile.

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Filling in the value we have for k, the downshore length BS is ...

BS = 1/√((5/3)² -1) = 1/√(16/9) = 3/4

The distance from point A to S is 4 miles -3/4 miles = 3 1/4 miles.