The lazy electrician.

ceptimus · Post by **ceptimus** » Tue Jul 20, 2004 4:12 pm

By a remarkable coincidence, the positions of Tommy's shed, garage and greenhouse lie at three of the corners of a 100-metre square; his house lies at the remaining corner.

Tommy has decided to run an electricity supply from his house to each of the outbuildings, none of which currently has a supply. Local planning regulations state that the cables must be buried underneath the ground, and Tommy hates digging trenches.

Tommy sits down and works out the minimum length for the trenches he will have to dig. Assuming that Tommy is extremely smart, what is his answer?

ratbag · Post by **ratbag** » Tue Jul 20, 2004 4:29 pm

I reckon
283 metres.....2 diagonals meeting in the middle gives access to all four corners of the square
Rat

Brown · Post by **Brown** » Tue Jul 20, 2004 9:57 pm

If my math is right, I get ... 273.205 meters.

Sundog · Post by **Sundog** » Tue Jul 20, 2004 10:02 pm

I get Rat's answer, 2 times the square root of 20,000. How did you cut it by 10 meters?

Brown · Post by **Brown** » Tue Jul 20, 2004 10:15 pm

It's a little hard to describe, but here goes: Instead of one central node, I have two central nodes, each node offset from the center of the square. Each "leg" stretching from the corner to a node leaves the corners at an angle of 60 degrees. (By contrast, with a single central node, each "leg" from the corner to the central node leaves at an angle of 45 degrees.) Each "leg" has a length of 57.735 meters, and the two nodes are 42.265 meters apart. 4*57.735 + 42.265 = 273.205 meters.

Looked at another way: If you have one central node, there are four lines leaving from it, with an angle of 90 degrees between any two neighboring lines. If you have the two nodes as I have described--or at least tried to describe--then each node has three lines leaving from it, with an angle of 120 degrees between any two lines.

Brown · Post by **Brown** » Tue Jul 20, 2004 10:23 pm

The pattern I tried to describe looks something like this:

Code: Select all

*...............*
..*...........*..
....*.......*....
......*****......
....*.......*....
..*...........*..
*...............*

[/color]

Sundog · Post by **Sundog** » Tue Jul 20, 2004 10:26 pm

Yes, I get it. Rather brilliant. I'm just trying to see how you arrived at that exact angle as the optimal solution.

ceptimus · Post by **ceptimus** » Wed Jul 21, 2004 11:11 am

Sundog wrote:Yes, I get it. Rather brilliant. I'm just trying to see how you arrived at that exact angle as the optimal solution.

You can use trial and error, or differentiate the formula for the length, and solve when the derivative equals zero.

A 'practical' way to see the solution is to make a long square wireframe prism and dip it in soapy water. The soap film, because of surface tension, will adopt the surface with the least area that bridges the four wires.

Brown got the correct answer. Well done Brown!

Sundog · Post by **Sundog** » Wed Jul 21, 2004 2:29 pm

ceptimus wrote:
Sundog wrote:Yes, I get it. Rather brilliant. I'm just trying to see how you arrived at that exact angle as the optimal solution.
You can use trial and error, or differentiate the formula for the length, and solve when the derivative equals zero.

A 'practical' way to see the solution is to make a long square wireframe prism and dip it in soapy water. The soap film, because of surface tension, will adopt the surface with the least area that bridges the four wires.

Brown got the correct answer. Well done Brown!

Ah well, I need to brush up on my calculus anyway. My son will be asking for help on his calc homework before long.

WildCat · Post by **WildCat** » Thu Jul 22, 2004 9:39 pm

[nitpick]
You're all wrong. He has to dig 300 meters (it could be a little less if you connect to the inside corners of the buildings of course) of trench, since all your solutions would require a buried junction box. AFAIK, there are no junction boxes for residential use that can be buried.
[/nitpick]

Sundog · Post by **Sundog** » Thu Jul 22, 2004 9:42 pm

WildCat wrote:[nitpick]
You're all wrong. He has to dig 300 meters (it could be a little less if you connect to the inside corners of the buildings of course) of trench, since all your solutions would require a buried junction box. AFAIK, there are no junction boxes for residential use that can be buried.
[/nitpick]

[nitpick-nitpick]Do the rules of the puzzle preclude an above-the-ground junction box?[/npnp]

ceptimus · Post by **ceptimus** » Thu Jul 22, 2004 9:59 pm

WildCat wrote:[nitpick]
You're all wrong. He has to dig 300 meters (it could be a little less if you connect to the inside corners of the buildings of course) of trench, since all your solutions would require a buried junction box. AFAIK, there are no junction boxes for residential use that can be buried.
[/nitpick]

If you're going to nitpick, get it right! :)

Having dug the trench, Tommy can easily run three separate wires through it: one to each of the outbuildings. I said that Tommy hates digging trenches, not that he is short of wire. ;)

WildCat · Post by **WildCat** » Fri Jul 23, 2004 2:38 am

ceptimus wrote:If you're going to nitpick, get it right! :)

Having dug the trench, Tommy can easily run three separate wires through it: one to each of the outbuildings. I said that Tommy hates digging trenches, not that he is short of wire. ;)

D'oh!