Best Answer superprismatic, 27 February 2014 - 08:56 PM

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Guest Message by DevFuse

Started by bonanova, Feb 25 2014 10:39 PM

Best Answer superprismatic, 27 February 2014 - 08:56 PM

Spoiler for that nice hint makes it look like

Go to the full post
6 replies to this topic

Posted 25 February 2014 - 10:39 PM

Here is a piece of plywood in the shape of an isosceles triangle.

The side lengths are 1, 1, sqrt(2) units.

Quick and dirty representation:

**A**

| \

| \

| \

| \

**B**--------------**C**

The angle at **B** is a right angle.

We'd like to cut this into two pieces of equal area.

There are many ways to do this with a single cut.

Which cut has the shortest distance?

*Vidi vici veni.*

Posted 26 February 2014 - 10:50 PM

Spoiler for cutting lines

Posted 27 February 2014 - 10:11 AM

Spoiler for cutting lines

With a careful reading of the OP can you do slightly better?

*Vidi vici veni.*

Posted 27 February 2014 - 08:56 PM Best Answer

Spoiler for that nice hint makes it look like

Posted 27 February 2014 - 09:01 PM

Kudos to both TSLF and SP.

*Vidi vici veni.*

Posted 27 February 2014 - 10:57 PM

@Bonanova or anyone else who may have an interest in this:

Do you have a proof that that circular arc is the shortest? Conceivably, a piece of a trigonometric curve, or exponential curve, or another conic section may be shorter. The possibilities are endless.

Posted 27 February 2014 - 11:41 PM

@Bonanova or anyone else who may have an interest in this:

Do you have a proof that that circular arc is the shortest? Conceivably, a piece of a trigonometric curve, or exponential curve, or another conic section may be shorter. The possibilities are endless.

Yes, it's called the isoperimetric problem, generally stated,

what is the curve of constant length that encloses the greatest area?

Equivalently, what shape of constant area has the smallest perimeter?

The answer is a circle. A 45^{o} arc can be reflected 7 times into a full circle.

Symmetry demands this answer, although rigorous proofs abound.

Here is a general discussion.

In three dimensions (think soap bubble) a sphere encloses more volume

than any other closed surface of the same area.

isoperimetric = iso (same) perimetric (perimeter).

*Vidi vici veni.*

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