Shortest distance problem

[rex21]

A stereo system is installed in a room with a rectangular floor measuring 12 feet by 10 feet and an 8 foot ceiling. There stereo is on the floor in one corner of the room and a speaker is on the floor in the opposite corner of the room. Your wire must run along the floor or walls (not through the air). What is the shortest length of wire you can use?

 

[HallsofIvy]

A stereo system is installed in a room with a rectangular floor measuring 12 feet by 10 feet and an 8 foot ceiling. There stereo is on the floor in one corner of the room and a speaker is on the floor in the opposite corner of the room. Your wire must run along the floor or walls (not through the air). What is the shortest length of wire you can use?

That's pretty straightforward. The 8 foot height is irrelevant because both stereo and speaker are on the floor. Run the wire from the stereo along one wall to the opposite side, then along the adjoining wall to the speaker. How long is each of those walls? What is the total length?

 

[rex21]

My mistake the speaker is actually on the ceiling in the opposite corner.

 

[Deleted member 4993]

A stereo system is installed in a room with a rectangular floor measuring 12 feet by 10 feet and an 8 foot ceiling. There stereo is on the floor in one corner of the room and a speaker is on the floor in the opposite corner of the room. Your wire must run along the floor or walls (not through the air). What is the shortest length of wire you can use?

It is still straight forward!

What is the length of a straight line between two points with co-ordinates (x₁, y₁, z₁) and (x₂, y₂, z₂)

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[HallsofIvy]

Reading it again, my solution was not even correct for the case where stereo and speaker are on the floor. I was thinking you had to run the wire along the wall. If you can run it straight across the floor, the distanc to the opposite corner, still on the floor, is given by the Pythagorean theorem- a diagonal of a rectangle with sides of length 12 and 10 is [itex]\sqrt{12^2+ 10^2}= \sqrt{244}= 2\sqrt{61}[/tex]. To run the wire up to a speaker on the ceiling, add the 8 foot height.

 

[lookagain]

Reading it again, my solution was not even correct for the case where stereo and speaker are on the floor.
I was thinking you had to run the wire along the wall. If you can run it straight across the floor, the distanc to the opposite corner,
still on the floor, is given by the Pythagorean theorem- a diagonal of a rectangle with sides of length 12 and 10 is
[itex]\sqrt{12^2+ 10^2}= \sqrt{244}= 2\sqrt{61}[/tex]. To run the wire up to a speaker on the ceiling,
add the 8 foot height.

That is not a correct solution. This is a variation of the "spider and the fly" puzzle.

Make an open box with the width, length, and height in the ratio of 10:12:8.

Fold down the sides of the box so that it is in a plane. Connect the corner (stereo) to the corner
where the speaker is with a line. Draw the line on the paper.

This forms a diagonal of a rectangle that is (10 + 8) units by 12 units. The minimum distance is

\(\displaystyle \sqrt{18^2 + 12^2} \ = \ \sqrt{468} \ \approx \ 21.6333 \ \ (units.)\)


Fold the paper back into an open box and see that the line is across the a portion of the floor
(but not the floor diagonal) and diagonally on a portion of a wall.


- - -

This distance is shorter than the last method/(intended solution) given by HallsofIvy.


The shortest distance is about 21.6333 feet.


Edit:

Subhotosh Khan stated after this post: "This is a classic fly walking on wall problem. "

Yes, I already stated to that effect at the top of this post.


Also, I deleted one of my posts.

 

[Deleted member 4993]

Subhotosh Khan,

rex21 mentioned that the wire will not run "through the air."

So, your straight-line distance between those 3-D coordinates will not apply, because the wire running along
some combination of the floor and walls makes it necessary to be longer than that straight-line distance
you mentioned.

I was not implying that cube diagonal is the answer. My reference was for generalized distance, piece-wise function. For some part, it is possible have x₁ = y₁ = x₂ = y₂ = 0 and for some part z₁ = z₂ =0.

This is a classic fly walking on wall problem. Since the student did not indicate what s/he knew, I started with most generalized formula - later see student's thought and guide towards the optimum.