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The Magic Cafe Forum Index » » Puzzle me this... » » How to paint a funnel (0 Likes) Printer Friendly Version

TomasB
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Imagine the function f(x)=1/x defined between 1 and infinity. Imagine it rotated around the x-axis to create a funnel with an opening with a diameter of 2.

What is the volume of this funnel?

What is the area of this funnel?

/Tomas
Daegs
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Oooh I love this one.

I really liked the proposal that our universe was shaped in this manor, which would give both the finite camp and infinite camp good answers and I think might explain some things...
Nir Dahan
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Tomas,

I assume no calculus is allowed...

Nir
TomasB
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Nir, solve it by any means you like. It's the result in this case that is interesting, not how you get there. As long as you do it right of course. Smile

/Tomas
leonard
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North Carolina
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Tomas,

I remember seeing a cartoon about this in High School. I believe it was called "Infinite Acres," and dealt with a character building a house as you describe. Seems he could never store enough paint to get the job done.

Mel Henrickson's name comes to mind (one of my math profs in college), although I think he had something to do with the cartoon presentation and not the problem itself.

I will give this a go later this week.

leonard

P.S. I look forward to Nir's non-calculus solution.
NCR
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Northfield Mount Hermon, MA
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Wow. I don't even know where to start...And I thought I was at least not bad at math...
TomasB
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NCR,

Big spoiler below...

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If you cut a disc at point x out of the funnel it will have the area pi*f(x)^2 and the circumferance 2pi*f(x). Multiply those with an infinitisemal thickness (dx) and you will get a local volume respective a local surface area. Sum those up from 1 to infinity and you will find the answers.

/Tomas
TomasB
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Hmmm, I'll just write a procedure to get the answer, but please ponder what the answer actually means. I've called the natural logarithm "ln".

V = integrate from 1 to inf(pi*f(x)^2 dx) = pi*integrate from 1 to inf(1/x^2 dx) = pi*[-1/x]:1:inf = pi*(-1/inf + 1/1) = pi

A = integrate from 1 to inf(2pi*f(x) dx) = 2pi*integrate from 1 to inf(1/x dx) = 2pi*[ln(x)]:1:inf = 2pi*(ln(inf) - ln(1)) = inf

In other words you have an object which you can't paint, since its surface is infinite, but you CAN fill it with paint.

/Tomas
Nir Dahan
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Tomas,

I think there are other interesting examples of surfaces you can paint (area is limited) but has infinite circumference (you can't trace it with a pen).
I believe the koch fractal is one of them but I may be wrong.

Nir
TomasB
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Nir, you are very right. A fractal border can be infinite but hold a finite area. I _think_ you can find borders that are fractal but still finite though, so they should be able to trace with a finite amount of ink but since you have to change direction of the pen an infinite number of times to draw it...

/Tomas
Nir Dahan
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Tomas,

along these lines, I remember years ago in my first year in uni the lecturer showed us a function which is continuous but can not be derived at any point.

Nir
TomasB
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Nir, there are many examples of such functions. Usually an infinite sum of sawtooth shapes that gets infinitely denser. Quite similar to the Koch fractal you wrote of, actually.

/Tomas
2hot
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Dude you didn't even give us the boundary for where the funnel ends
TomasB
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Ok, 2hot, you may give the answer where the boundary is [1,t] where t is a real number and t > 1.

/Tomas
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