A stereo system is being installed in a room with a rectangular floor measuring 15 feet by 11 feet and a 10-foot ceiling. The stereo amplifier is on the floor in one corner of the room. A speaker is at the ceiling in the opposite corner of the room. You must run a wire from the amplifier to the speaker, and the wire must run along the floor or walls. What is the shortest length of wire you can use for the connection?

If we want to go from one point in space to another one, the shortest possible distance is through a line. But if we are not allowed to, we must try to use as many direct paths as possible to minimize the distance.

In our case, the minimum distance will be going right from one corner to the other through the air, but we must use walls and floor. So our choices are limited to shorten the distance as much as possible by taking paths from corners whenever it's possible.

Starting from the corner where the stereo system is located, we must try to reach the other corner by using one of these options

Cross the floor by its diagonal and then go right to the other corner in the ceiling
Cross one wall by its diagonal and then go to the other corner. It can be done for each adjacent wall to the stereo system.

If we try the first option first, the diagonal of the floor measures

[tex]\displaystyle \sqrt{15^2+11^2}[/tex]

And now we reach the other corner, traveling a total distance of

[tex]\displaystyle L_1\sqrt{15^2+11^2}+10[/tex]

[tex]\displaystyle L_1=28.60\ {ft}[/tex]

Trying one of the adjacent walls

[tex]\displaystyle L_2=\sqrt{15^2+10^2}+11[/tex]

[tex]\displaystyle L_2=29.028\ feet[/tex]

And the final option is

[tex]\displaystyle L_3=\sqrt{10^2+11^2}+15\ feet[/tex]

[tex]\displaystyle L_3=29,87\ feet[/tex]

We can see the first is the shortest distance, so to reach the speaker we must cross the floor by its diagonal and the go up to the speaker

Note: Each of the options analyzed have a identical counterpart by taking the opposite side first. For example, the path floor-wall could have been done as wall-ceiling, with identical results