Good problem but uncertain WHY

[Robokapp]

Okay... i did solve it, but i wonder why the answer is what it is.


"a string 10 m long is cut in 2 so that one piece forms one square and one piece forms an equlateral triangle". HOw do you cut it so the total area obtained is minimalized...and part b. maximalized?

For maximalized i got that everything must go into the square, which makes sense...because at a given perimeter the area is bigger in a polygon with a larger number of sides, of course. But the minimum comes out to be at about 4.45 (if i remember right) meters into the square. Now...why isn't everything going into the triangle? I got some equations but I don't know how to write them in the 'cool way' so i won't even bother.

 

[mezarashi]

That's the whole purpose of mathematics. When we have a situation that's a bit far too complex to analyze intuitively at all steps, we let the math do the work.

If there is any explanation at all, having more shapes will definitely waste more string. Each new shape requires some 'start-up'. Let's take an example of two circles versus one circle.

Let us choose a rope of length C, then it's radius and area are:

$$R = \frac{C}{2\pi}, A = \pi (\frac{C}{2\pi})^2 = \frac{1}{4} \frac{C^2}{\pi}$$

Now if we cut this C in half and distribute it among two circles instead:

$$R = \frac{C}{4\pi}, A = 2 \times \pi (\frac{C}{4\pi})^2 = 2 \times \frac{1}{16} \frac{C^2}{\pi} = \frac{1}{8} \frac{C^2}{\pi}$$

 

[Robokapp]

That is one beautiful explanation. That was truly great. Thank you. It makes sense now.

Well, I got one problem for you that pretty much proves your point. It was shown to me by my dad. Common sense can not help you...it's too out of proportion for someone's mind to understand it without thinking of a few formulas.

"if you increase the equator of Earth by 1 meter and you place the new circle so the two circles are circumcentrics, will a cat be eable to pass in between them and come out alive?

Circles are in same plane...no 3-d stuff.

 

[HallsofIvy]

"if you increase the equator of Earth by 1 meter and you place the new circle so the two circles are circumcentrics, will a cat be eable to pass in between them and come out alive?

Well, not "increase the equator of the earth", but make a chain 1 meter longer than the circuference of the Earth and put it around the Earth so that its center is at the center of the earth.

If the radius of the Earth is R, then its circumference is $2\pi R$. That means the length of the chain is $2\pi R+ 1$ and that is the circumference of a circle with radius $\frac{2\pi R+ 1}{2\pi}= R+\frac{1}{2\pi}$. That is, it will be [itex]\frac{1}{2\pi}[/tex] meter off the earth. That is about 0.16 m or 16 cm. How tall is your cat?

 

[Robokapp]

Well, not "increase the equator of the earth", but make a chain 1 meter longer than the circuference of the Earth and put it around the Earth so that its center is at the center of the earth.
If the radius of the Earth is R, then its circumference is $2\pi R$. That means the length of the chain is $2\pi R+ 1$ and that is the circumference of a circle with radius $\frac{2\pi R+ 1}{2\pi}= R+\frac{1}{2\pi}$. That is, it will be [itex]\frac{1}{2\pi}[/tex] meter off the earth. That is about 0.16 m or 16 cm. How tall is your cat?

:D it's smaller than that. Don't forget they can crawl.

But what I'm trying to point out is even if it's a simple problem a 10th grader can solve with basic geometry, it's impossible to picture in your head. The rings are too big to comprehend the space in between them as anything but "tiny" which you don't know how much it means compared to real life sizes.

I thought it's a good problem. It is similar to your 'starting point" thing. At least in my mind it makes sense and connection.

WEll, thank you.
~Robokapp