Wikipedia:Reference desk/Archives/Mathematics/2007 January 27

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[edit] January 27

[edit] Polygon

I'm sorry for my bad English. I have a polygon with 2007 corners and the numbers 1 to 4014. Now you must place the numbers on each corner and on each side. (Every number can only used once). The sum of two corners and "their side" must be everywhere on the polygon the same. What should be done? Are there any formulas for understanding this?

In German

Auch auf Deutsch, da hoffentlich so jemand etwas mehr versteht. Ich habe ein 2007-Eck und die Ziffern 1 bis 4014. Alle Ziffern müssen auf den Ecken und den Seiten verteilt werden (jede Ziffer darf nur einmal benutzt werden). Jede Seite muss nun mit seinen Ecken eine Summe haben, die überall gleich ist. Was soll man tun? Gibt es für die Anordnung eine Formel? --85.178.18.172 13:29, 27 January 2007 (UTC)

Your English is pretty good; I copy-edited the above (you can look in the page history to see the corrections). There's plenty of solutions to the problem, but there's one particularly elegant solution that can be described in only a few lines. Try experimenting with smaller polygons; a triangle, then a pentagon (to see the form of the general solution), then a heptagon (to check that it works). The solution only works on odd-sided polygons; why? To help you on your way: what does each sum have to come to if you place the numbers 1-2007 on the corners and 2008-4014 on the sides? –EdC 14:59, 27 January 2007 (UTC)

I'm sorry for my bad German. Sie können diese Web site besichtigen und Ihre Frage stellen:

hier

--Ķĩřβȳ♥ŤįɱéØ 03:09, 28 January 2007 (UTC)

PLEASE DO NOT GIVE ANY HINTS. This is a problem from an on-going competition, the anonymous user is trying to cheat.--80.136.135.24 20:44, 30 January 2007 (UTC)

[edit] Free fall

In free fall, how far will you have travelled after t seconds? Thanks in advance, Jack Daw 15:43, 27 January 2007 (UTC)

Depends on (i) your initial velocity and (ii) the effect of air resistance (unless this can be neglected). Obviously also depends on gravity, but I am assuming you mean free fall near the surface of the Earth or in low Earth orbit. Gandalf61 16:10, 27 January 2007 (UTC)
Yes, free fall near the surface of the Earth. Initial velocity, well 0 m/s I guess. Jack Daw 21:57, 27 January 2007 (UTC)
Cautionary joke:
  • Two hot-air baloonists are drifting lost in a deep canyon. In desperation, one shouts over the side hoping someone will hear: "Where are we?" Several minutes later they hear: "You're in a balloon!" The other one says, "That was a mathematician." Asked how he came to that conclusion, he replied,
    1. "The answer took a long time."
    2. "It was perfectly correct."
    3. "It was absolutely useless."
This question properly belongs on the science reference desk, but since you're here: According to Newton, the force due to gravity for two bodies of (gravitational) masses M and m will be G times Mm/d2, where d is the distance between them, and where G is the gravitational constant. Newton also says force equals mass times acceleration, F = ma. Notice that the acceleration, a = F/m = GM/d2, is independent of m. We treat the Earth's mass, M, as concentrated at a point at its center, and treat the distance as the Earth's radius with negligible deviation. Under this standard approximation, the acceleration is essentially constant. Thus the velocity at time t is given by the product of this constant acceleration, a, and time
v(t) = a t , \,\!
and the distance at time t is given by the integral of velocity.
\begin{align} \operatorname{dist}(t) &{}= \int_0^t v(\tau) \, d\tau \\ &{}= \int_0^t a \tau \, d\tau \\ &{}= \tfrac12 a t^2 \end{align}
We can estimate a from the known values of Earth's mass and radius, or use actual physical measurements (very un-mathematical!). The experimental results are fascinating, as the GRACE satellite data illustrates. Still, the variations are negligible for most purposes, so we can use the standard value 9.80665 m/s2, or approximately 32 feet per second per second.
Well, I did warn you! --KSmrqT 23:13, 27 January 2007 (UTC)

[edit] math question

Where does absolute value fall in order of operation?

State the multiplication property of -1. —The preceding unsigned comment was added by Ladyjo13 (talk • contribs) 16:31, 27 January 2007 (UTC).

The absolute value is equivalent to parentheses )( in order of operation.

And as to your second question, -1 has many multiplication properties. You should do your own homework and ask your teacher what property.--Ķĩřβȳ♥ŤįɱéØ 03:13, 28 January 2007 (UTC)

As a start for your homework read this two articles Absolute value & Negative and non-negative numbers. Mathmo Talk 00:32, 29 January 2007 (UTC)

[edit] Proof that 1 = 2

I'm looking for proofs that 1 = 2 or that 0 = 1. I've found the well-known proofs that divide by zero, or that equate the positive and negative square roots of 1, and I've developed proofs that take the natural log of a complex number, and that equate infinities. Are there any others out there? --Carnildo 22:48, 27 January 2007 (UTC)

Invalid proof has a lot of them. --Spoon! 23:15, 27 January 2007 (UTC)