Math geeks?

JesusIsLove

JesusIsLove

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May 24, 2009
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I think that I picked up on this in an integral calculus class. It has bothered me ever since. It is so anti-intuitive, and I still can't understand how it is possible.

problem:

You have a length of fence wire.. say 20 feet.
You are going to build a dog pen or something.
You try out a few different shapes.. square, circle, rectangle.
You discover that the area enclosed is different depending on the
shape of the enclosure. How is this possible?

square 5x5x5x5 = 25 square feet
rectangle 3x3x7x7 = 21 square feet
circle c=20 = ~32. square feet

same length of fence surrounding each pen (20 ft)
wtf?
 


Square = 5+5+5+5 = 20ft
Rectangle = 4+4+6+6 = 20ft
Circle : 20 ft = 2 pie squared

.
.
Edit: Wait, I see that this is "area", not "perimeter".
 
So if this is about area, wouldn't it make sense that if the perimeter was the same (20ft), the area would stay the same too? ...or are you asking a different q?
 
Garrett said:
So if this is about area, wouldn't it make sense that if the perimeter was the same (20ft), the area would stay the same too? ...or are you asking a different q?
Click to expand...

Yeah.. that would make sense, but it doesn't turn out that way. That is what is weird.
 
Imagine if you had a circle (envision it as a circular birthday cake that you are looking down at from above) and you take a slice out of it. You will have increased the perimeter of the shape, but decreased the area (lets assume your slice goes all the way to the midpoint of the circle, and that it's not a fat slice, so we know for a fact that the two sides of the slice are longer than the arc of the circle that was there before.) if you were to replace the slice, you will have restored the circle, DECREASED the perimeter, and INCREASED the area.
 
well, OP you forgot you could optimize the non-square rectangle by doing 4x6 instead of 7x3 - it'd come out as a 24sq foot area. it seems counter intuitive, but it's just based on the shape itself. remember a circle has an infinite number of sides, whereas the square and rectangle only have 4 each. this isn't really integral calculus, this is differentiation. you find the optimum sides based on the amount of feet of fencing available by taking the derivative of your area function, and setting the derivative equal to zero. at this point, you have the point at which your area is optimum.
 
I don't see what the big fuzz is about. Why do you even assume that the areals will be the same for every type of shape as long as the perimeter is the same, to begin with?
 
papajohn56 said:
well, OP you forgot you could optimize the non-square rectangle by doing 4x6 instead of 7x3 - it'd come out as a 24sq foot area. it seems counter intuitive, but it's just based on the shape itself. remember a circle has an infinite number of sides, whereas the square and rectangle only have 4 each. this isn't really integral calculus, this is differentiation. you find the optimum sides based on the amount of feet of fencing available by taking the derivative of your area function, and setting the derivative equal to zero. at this point, you have the point at which your area is optimum.
Click to expand...

It's worth noting that the closer a rectangle comes to becoming a square, the more square footage you attain:

6 x 4 = 24
5.5 x 4.5 = 24.75
5.1 x 4.9 = 24.99
5.01 x 4.99 = 24.9999

and so on. This applies to triangles as well (equilateral for the sake of conversation, but it works with scalene and isosceles as well):

6.5 x 6.5 x 7 = 18.97
6.6 x 6.6 x 6.8 = 19.233
6.6666 x 6.6666 x 6.6666 = 19.2446

Notice that you can't get as much space out of a triangle as you can a square :p
 
Well it's just not intuitive to you because you are thinking about it wrong. Let's use surface area and volume and think in 3d instead. (because the same idea applies same surface area != same volume)

Now think about blowing up a balloon. Would you expect it to take any other shape but a sphere? Most people would intuitively agree it makes perfect sense it forms a sphere. This is because the balloon doesn't want to stretch so in order to stretch as little as possible it takes a shape to minimize surface area for a given volume, a sphere. This is also why bubbles are spheres.

Plenty of things in math are very odd and non-intuitive but I never thought of this is as one of them. Here are a few that I always liked to share with my students that have to do with probability.

The HIV test is 99.89% accurate it only gives false positives .11% of the time. So if you get a positive on your FIRST HIV test in America what are your chances of having HIV? About 58%. They have to do another test to have a better idea.

If you have a random group of 67 people in a room the chances that you CAN NOT find 2 of them with the same birthday, well you have a better chance of winning the California state lottery.

And for you graduate level math geeks that want something really baffling. If you accept the fact that in any finite group of unique natural numbers there is a greatest number. Well you can prove with that fact that if you take two spheres and chop them up in the right way you can put them back together to make two new spheres of the same size and density. Now that's something I still can't wrap my brain around even though I understand the proof in theory.
 
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