16 Jun 10
1 edit
One more:
A flexible cable was hung across a chasm between two points that were exactly 1 km apart and at the same elevation. During the cool night the cable length was calculated to contract by .2 meters. The cable dip was actually measured to decrease by .2 meters. What is the length of the cable after cooling?
17 Jun 10
Originally posted by clandarkfireThe cable follows a function F(x) = cosh(x) (I have no desire to explain why)
One more:
A flexible cable was hung across a chasm between two points that were exactly 1 km apart and at the same elevation. During the cool night the cable length was calculated to contract by .2 meters. The cable dip was actually measured to decrease by .2 meters. What is the length of the cable after cooling?
The lenght of the cable then is L = INTEGRAL(SQRT(1+F'(x)^2))dx from x=0 to x= 1000m.
L = INTEGRAL(SQRT(1+sinh(x)^2))dx = INTEGRAL(cosh(x))dx = sinh(1000)
The length becomes sinh(1000)-0.2, while the height of the dip becomes cosh(500)+0.2
How to proceed is too much for me to think through right now :p
17 Jun 10
Originally posted by clandarkfirehttp://calculuslab.deltacollege.edu/ODE/7-A-1/7-A-1-h.html
That's a good start! Anyone want to help out the master? My first thought was that it was a parabolic curve which would make it a lot easier. It's actually a catenary, but the length can still be found with the help of a little calculus. 🙂
Seems like too much math for me.
17 Jun 10
Deriving the formula is the interesting part! Calculus of variations, with a constraint (minimise the potential energy subject to the length being fixed).
It is, admittedly, not basic maths. I think it was in my final year of an undergraduate maths degree where I first had to solve that particular problem.
17 Jun 10
Originally posted by mtthwYeah, I don't remember if DiffEq was 2nd or 3rd year of my engineering degree, but I've forgotten most of it by now anyway.
Deriving the formula is the interesting part! Calculus of variations, with a constraint (minimise the potential energy subject to the length being fixed).
It is, admittedly, not basic maths. I think it was in my final year of an undergraduate maths degree where I first had to solve that particular problem.
17 Jun 10
Originally posted by mtthwSo what set of degrees did you end up with?
Deriving the formula is the interesting part! Calculus of variations, with a constraint (minimise the potential energy subject to the length being fixed).
It is, admittedly, not basic maths. I think it was in my final year of an undergraduate maths degree where I first had to solve that particular problem.
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18 Jun 10
I'm puzzled about the lines:
>>The lenght of the cable then is L = INTEGRAL(SQRT(1+F'(x)^2))dx from x=0 to x= 1000m.
>>L = INTEGRAL(SQRT(1+sinh(x)^2))dx = INTEGRAL(cosh(x))dx = sinh(1000)
The cable length is not given in the question, and nor is the dip, so it is possible the cable is:
a) Taught, with a length of 1000
b) Loose, with a longer length
I would have therefore expected a function at this stage which had a minimum of 1000 and a maximum of +infinity, the equation given has a constant value though - what am I missing?
18 Jun 10
Originally posted by iamatigerIf the cable is taught, the dip can't rise by 0.2 meters after cooling.
I'm puzzled about the lines:
[b]>>The lenght of the cable then is L = INTEGRAL(SQRT(1+F'(x)^2))dx from x=0 to x= 1000m.
>>L = INTEGRAL(SQRT(1+sinh(x)^2))dx = INTEGRAL(cosh(x))dx = sinh(1000)
The cable length is not given in the question, and nor is the dip, so it is possible the cable is:
a) Taught, with a length of 1000
b) Loose, with a longer ...[text shortened]... nd a maximum of +infinity, the equation given has a constant value though - what am I missing?[/b]
If it's longer then the shape of the cable is a caternary, described by cosh(x). There is a whole derivation of this on the web, but it was too much to copy it here 🙂
The length between x=0 and x=1000 is of a given line F(X) always given by the integral in the first quotes sentence.
Combining the two give the second quoted sentence.
21 Jun 10
Originally posted by forkedknightThis is (in, for example, set theory) incorrect.
There are only two magnitudes of infinity. Countably infinite, and uncountably infinite.
There are infinitely many magnitudes of infinity. They are often denoted by the alephs (wiki `cardinal numbers'😉.
Countably infinite means you can count the set - that you can put the set in a list indexed by the natural numbers. Uncountably means you can't, but there are lots of levels of `can't'. For example, let X be any set. Then the powerset of X has cardinality strictly greater than that of X, |P(X)|>|X| (there exists an injection from X into P(X) but there does not exist an injection from P(X) to X). Thus, if we start with N, the natural numbers, we have a chain,
|N|<|P(N)|<|P(P(N))|<...
It is known that |P(N)|=|R|, where R is the real numbers. We are then left with the rather natural question...does there exists a set X such that |N|<|X|<|R|? This question is the `continuum hypothesis', and it's answer is...well, complicated.
21 Jun 10
2 edits
Originally posted by clandarkfireThere are always problems with integer sequences, namely that there is never a unique answer.
1). 1,2,5,14: find the next term and a formula for the nth term.
In fact, there is a perfectly reasonably reason why your sequence should go,
1, 2, 5, 14, 33, 66, 117, 190, 289, 418...
(You gave the results for x=1, 2, 3, 4 of the polynomial (2/3)x^3-3x^2+(16/3)x-2).
The classic example of a sequence with two endings is,
1, 2, 4, 8, 16. What is the next numbers (hint: it is not 32).
21 Jun 10
Originally posted by FabianFnasFor the first point, yes, this is correct (if you take your ordering to be cardinality, as |N|<|R|, look up Cantor's Diagonal Argument).
You, Swlabr, seem to know some about transfinite numbers...
Let N be the number of natural numbers.
Let R be the number of real numbers.
Is it correct to say that R > N ?
Is there any X such as N < X < R ?
I referred in my post which you just replied to to this problem. The answer to the problem is...there is no solution. You can assume that there is such a set, or you can assume that there is no set. It makes no difference to anything. In this way, it is similar to the axiom of choice.