Where should the junction box be placed to minimize the length of wire needed in the following problem?Two isolated farms are situated 12 km apart on a straight country road that runs parallel to...

Where should the junction box be placed to minimize the length of wire needed in the following problem?

Two isolated farms are situated 12 km apart on a straight country road that runs parallel to the main highway 20 km away. The power company decides to run a wire from the highway to a junction box, and from there, wires of equal length to the two houses.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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Two isolated farms are situated 12 km apart on a straight country road that runs parallel to the main highway 20 km away. The power company decides to run a wire from the highway to a junction box, and from there, wires of equal length to the two houses.

Let the box be on a line that is a perpendicular bisector of the line joining them and at a distance x from the highway. The length of wire needed is D = `x + 2*sqrt(6^2 + (20 - x)^2)`

=> D = `x + 2*sqrt(36 + 400 + x^2 - 40x)`

=> D = `x + 2*sqrt(x^2 - 40 x + 436)`

To minimize D, solve D' = 0

=> `1 + (1/sqrt(x^2 - 40 x + 436))*(2x - 40) = 0`

=> `2x - 40 = -sqrt(x^2 - 40 x + 436)`

=> `4x^2 + 1600 - 160x = x^2 - 40 x + 436`

=>` 3x^2 + 1164 - 120x = 0`

`x = 20 - 2*sqrt 3` (the other root can be ignored as it is greater than 20)

`x ~~ 16.53`

The junction should be placed approximately 16.53 km from the highway.

samandcristin's profile pic

samandcristin | Student, Undergraduate | eNotes Newbie

Posted on

My friend and I have the same question.. Where did the x + 2 come from in the original equation?

nerdynikki's profile pic

nerdynikki | Student, Undergraduate | (Level 3) eNoter

Posted on

^^ thanks for your answer. I actually figured out on my own and I did it right! thank you so much!

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