Volumes (2 Viewers)

[artosis]

i THINK i got the first part.
is it:

<a href="http://www.codecogs.com/eqnedit.php?latex=\frac{20-DC}{10} = \frac{x}{100}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\frac{20-DC}{10} = \frac{x}{100}" title="\frac{20-DC}{10} = \frac{x}{100}" /></a>

<a href="http://www.codecogs.com/eqnedit.php?latex=\text{Therefore} \\ DC = 20 - \frac{x}{10}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\text{Therefore} \\ DC = 20 - \frac{x}{10}" title="\text{Therefore} \\ DC = 20 - \frac{x}{10}" /></a>

And I have no clue about the next part.

 

[bleakarcher]

these problemz are quite interesting to be honest. i havent covered this part of volumez but i will soon. dont count on me on this. the question basically tells to take the limiting sum of the trapezoidal cross-sections from x=0 to 100 as the showroom is of length 100m. to do this you must find the area of the trapezium in terms of x. we have already the side length DC in terms of x and so once we find the side AB in terms of x, we may use the formula A=(1/2)h[a+b] in order to find the area of the trapezium. if u notice on the top of the 'solid' are two similar triangles the larger of base length 100m and height 20m and the smaller of base (100-x) and height AB. in similar triangles, corresponding sides are in the same ratio, hence:
AB/20=(100-x)/100
100AB=20(100-x)
AB=1/5[100-x]
Let the area of the trapezoid be A(x).
Now, A(x)=(1/2)[20][AB+DC]=10[20-(x/10)+20-(1/5)x]=10[40-(3/10)x]=400-3x
Let the thickness of the trapeziodal cross-section at some distance x from the square end of the showroom be delta x.
Now, V=lim[delta x approaches 0][sum from x=0 to x=100 of A(x)*delta(x)]=integral[400-3x] dx from x=0 to x=100=[400x-(3/2)x^2] from x=0 to x=100
V=400*100-(3/2)*100^2=25000m^3
im not sure about this but i gave it a go lol

 

[largarithmic]

By using similar triangles with the triangles on the top and AB, you get


Hence

Then taking a slice near x, you have:



So then integrating to find the volume (yes I am not doing any of this limiting sum bullshit, see the other thread about volumes)



which is what bleakarcher did, except without the limiting sum argument grrrr



Also speaking of this threads title, did you know volume doesnt actually exist?

 

[hup]

Also speaking of this threads title, did you know volume doesnt actually exist?

lol
the cylindrical shells slicing annulus ones are the imaginary ones
but ones like this are 'real'

 

[largarithmic]

lol
the cylindrical shells slicing annulus ones are the imaginary ones
but ones like this are 'real'

Nah I meant that legitimately, volume is not actually a property of 3 dimensional space.

 

[K4M1N3]

Nah I meant that legitimately, volume is not actually a property of 3 dimensional space.

Yeah but capacity just sounds stupid.

 

[largarithmic]

Yeah but capacity just sounds stupid.

Oh capacity doesn't exist either. What I mean to say is, there's actually no volume-like quantity (doesn't matter what you name it, volume, capacity, etc) in three dimensional space. It just doesn't exist.

 

[seanieg89]

Oh capacity doesn't exist either. What I mean to say is, there's actually no volume-like quantity (doesn't matter what you name it, volume, capacity, etc) in three dimensional space. It just doesn't exist.

What exactly do you mean by this? There certainly exists something called a measure on R^3, which satisfies all the reasonable properties you might expect of a "volume". Or are you referring to the fact that in the HSC course, you calculate "volumes" without them being properly defined?

 

[largarithmic]

What exactly do you mean by this? There certainly exists something called a measure on R^3, which satisfies all the reasonable properties you might expect of a "volume". Or are you referring to the fact that in the HSC course, you calculate "volumes" without them being properly defined?

I haven't done any university mathematics but I'm pretty sure a volume measure doesn't exist on R^3, even though an area measure exists on R^2. See http://en.wikipedia.org/wiki/Banach–Tarski_paradox.

 

[seanieg89]

Okay, i misunderstood you then . Where that 'paradox' comes from is the fact that it is impossible to define a measure satisfying all the desirable properties on the entire power set of R^3. My point is that there still exists a nice measure defined on the class of sets called Lebesgue measurable, which is such a large class that it encompasses all the solids dealt with in high school. Its actually quite tricky to construct a set in R^3 which is not Lebesgue measurable.

 

[largarithmic]

Okay, i misunderstood you then . Where that 'paradox' comes from is the fact that it is impossible to define a measure satisfying all the desirable properties on the entire power set of R^3. My point is that there still exists a nice measure defined on the class of sets called Lebesgue measurable, which is such a large class that it encompasses all the solids dealt with in high school. Its actually quite tricky to construct a set in R^3 which is not Lebesgue measurable.

Do you know if you can do it without the axiom of choice? The only counterexample I've seen in detail is Banach-Tarski at a short talk a few months ago, and the guy giving the talk kinda skipped over the axiom of choice part. If you weren't to accept the axiom of choice, would there still exists non Lebesgue measurable sets? As in could you prove their existence either nonconstructively, or provide a construction without the axiom of choice (and does the proof of Banach-Tarski actually rely on the axiom of choice, I assume it must because it would have to be constructive)?

 

[seanieg89]

Yes the proof of the Banach-Tarski Paradox relies crucially on AC.
I know that you cannot construct non-Lebesgue measurable sets without AC, as for non-constructive proofs of their existence I am not too sure. In any case the vast majority of mathematicians accept AC since there are several very basic results that cannot be proven without: eg every vector space has a basis.

By the way, am very impressed with you mathematical maturity, its good to see people with an obvious passion for it

 

[largarithmic]

Yes the proof of the Banach-Tarski Paradox relies crucially on AC.
I know that you cannot construct non-Lebesgue measurable sets without AC, as for non-constructive proofs of their existence I am not too sure. In any case the vast majority of mathematicians accept AC since there are several very basic results that cannot be proven without: eg every vector space has a basis.

I did a bit of wikipedia-ing and found this; http://en.wikipedia.org/wiki/Solovay's_model. Apparently you need an 'inaccessible cardinal' though (and I'm still not sure what this means and any meaningful explanation is probably far beyond what understanding of set theory I might have), and checking that, http://mathworld.wolfram.com/LebesgueMeasurabilityProblem.html, apparently it's also necessary to the proof. So I guess it completely hinges on choice.

By the way, am very impressed with you mathematical maturity, its good to see people with an obvious passion for it

Thanks lol, maths is pretty cool ^^ What are you doing for your honours thesis?

 

[seanieg89]

Hey yeah, and simply replacing ZF with another set theory in order to avoid invoking AC feels like cheating. Its pretty easy to get bogged down in these sort of things haha...set theorists are crazy.

I'm studying these highly symmetric functions on hyperbolic space sometimes known as automorphic forms (not the wiki entry), the harmonic analysis of the hyperbolic plane leads to a study of their spectral theory which has important applications to number theory. Its good because its a nice blend of several interesting fields of mathematics.

 

[largarithmic]

Hey yeah, and simply replacing ZF with another set theory in order to avoid invoking AC feels like cheating. Its pretty easy to get bogged down in these sort of things haha...set theorists are crazy.

I'm studying these highly symmetric functions on hyperbolic space sometimes known as automorphic forms (not the wiki entry), the harmonic analysis of the hyperbolic plane leads to a study of their spectral theory which has important applications to number theory. Its good because its a nice blend of several interesting fields of mathematics.

It's not really cheating lol, but yeah it feels like it. Its interesting to know that you absolutely need choice to find nonmeasurable sets. And this isn't really set theory is it? I was under impression Banach-Tarski and other such things were more about group theory, in particular rotation groups

I vaguely know what hyperbolic space is but otherwise I have no idea what any of that stuff is (apart from number theory lol). I'm planning to basically do maths at uni, hopefully do a phd. I dont really know anything about university maths though apart from vaguely there's a division between algebra analysis topology with some crossover fields.

 

[seanieg89]

BT is more about group theory yes, but the necessity of AC for constructing nonmeasurable sets feels more like set theory. The group theory comes into play when making the statement about moving the finite number of pieces of the sphere to reform two spheres, it has to do with the group of rotations in n>2 dimensional space. (though nonmeasurable sets themselves exist in every positive dimension of Euclidean space).

Sweet, yeah I did the same thing. Am applying to some universities in the US now to start a PhD next year