A power line is to be constructed from a power station at point A to an island a

Question

A power line is to be constructed from a power station at point A to an island at point C, which is 2 mi directly out in the water from a point B on the shore. Point B is 8 mi downshore from the power station at A. It costs S$3200 per mile to lay the power line under water and $2000 per mie to lay the line under ground. At what point S downshore from A should the line come to the shore in order to minimize cost? Note that S could very well be B or A. (Hint: The length sismiles from A. Round to two decimal places as needed.)

Explanation / Answer

Solution:

Cost function C(x) is given by

=> C(x) = 3200 sqrt( 4+x^2) + 2000(8-x)

=> C(x) = 3200 sqrt(4 + x^2) + 16000 - 2000x

=> C'(x) = 3200x(4 + x^2)^(-0.5) -2000

putting C'(x) = 0

=> 2000 = 3200 x / sqrt (4 + x^2)

=> 5/8 = x / sqrt (4 + x^2)

=> 8/5 x = sqrt ( 4 + x^2 )

=> 64 / 25 x^2 = 4 + x^2

=> 39 / 25 x^2 = 4

=> 39 / 100 x^2 = 1

=> x^2 = 100/39

=> x = 1.60 mi (approx)

Distance of S from A = 8-1.6

=> distance = 6.4 mi from A

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