The junction box is at an equal distance from the two farms. If a line perpendicular to the country road is drawn between the farms, the junction box is on it. Let the junction box be x km from the highway. It is 20 - x km from the country road.

The length of wire from the junction box to each of the farms is `sqrt((20 - x)^2 + 36)` . The length of the wire from the highway to the junction box is x. The total length of the wire required is `x + 2*sqrt((20 - x)^2 + 36)`

This has to be minimized. The derivative of `x + 2*sqrt((20 - x)^2 + 36)` is `1 + (2*(1/2)*2*(20 - x)*-1)/sqrt((20 - x)^2 + 36)` . Equating this to 0 and solving for x.

=> `1 + (2*(1/2)*2*(20 - x)*-1)/sqrt((20 - x)^2 + 36)` = 0

=> `-2*(20 - x) = -1*sqrt((20 - x)^2 + 36)`

=> `40 - 2x = sqrt((20 - x)^2 + 36)`

=> 4x^2 + 1600 - 160x = (20 - x)^2 + 36

=> 4x^2 + 1600 - 160x = 400 + x^2 - 40x + 36

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

The equation has two roots `20 + 2*sqrt 3 and 20 - 2*sqrt 3` . The distance cannot be greater than 20.

**The junction box should be placed `20 - 2*sqrt 3` km from the
highway.**

## See eNotes Ad-Free

Start your **48-hour free trial** to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Already a member? Log in here.