Wikipedia:Reference desk/Archives/Mathematics/2008 December 3

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December 3

Trapezoid

Hello. The short parallel base of a trapezoid and its two sides intersecting the bases are each 2 units long. What is the angle x between the long base and one of the sides?

Draw the height perpendicular to the long base intersecting one of the ends of the short base. Let the long base, b₂, be 2 + 2c.

Let $\sin x={\frac {h}{2}}$ and $\cos x={\frac {c}{2}}$ (equation 1).

The area is h(2 + c) = 5 or $h={\frac {5}{2+c}}$ (equation 2). Using Pythagorean identity, h² + c² = 4 (equation 3). Substitute (2) into (3). If ${\frac {25}{(2+c)^{2}}}+c^{2}=4$, then (2 + c)²(2 - c) - 25 = 0. How do I solve for c to substitute into (1)? Thanks in advance. --Mayfare (talk) 02:22, 3 December 2008 (UTC)

You are getting close. You can continue manipulating the equation until you obtain a quartic equation and then attempt to find its roots. If you are unable to factor the equation yourself, you can use a [quartic formula solver] such as [this one]. Note there may be multiple solutions, both real and imaginary. -- Tcncv (talk) 07:16, 3 December 2008 (UTC)

But, as far as I can tell, only one of the four solutions to the quartic gives a valid solution - c obviously cannot be imaginary or negative, but there's also an upper bound on it that is related to the triangle inequality. Confusing Manifestation(Say hi!) 22:04, 3 December 2008 (UTC)

The equation should have been (2 + c)³(2 − c) − 25 = 0. This equation has two positive solutions, c ~ 1.31166095 and c ~ 0.63764893, and two nonreal solutions c ~ −2.97465494 + 1.38278607i and c ~ −2.97465494 − 1.38278607i. I used the J (programming language): p.(25-2&-*3^~2&+)t.i.5 . Bo Jacoby (talk) 23:48, 3 December 2008 (UTC).

You beat me too it, but here's part of what I had: To visualize, consider a square (c = 0) with area 4 square-units. As c increases, the long parallel side increases and height decreases. However, for small values of c, the increased width contributes more to the area than the decreased height takes away. Eventually the area reaches and surpasses 5 square-units for some value of c (solution 1). As c continues to increase, the decreased height becomes a greater factor, and eventually area decreases and again passes 5 square-units (solution 2). -- Tcncv (talk) 00:23, 4 December 2008 (UTC)

Ratio Proportion Problem

I had a question about this problenm I'm working on and it says that a farmer wants to plant 896 acres of alfalfa and wheat in the ratio 3 acres of alfalfa for every 4 acres of wheat. It asking how many acres should be planted in each crop? I don't know where to begin in this problem and every step I have taken I think might be wrong. Could anyone help? I love math!--70.250.226.169 (talk) 02:49, 3 December 2008 (UTC)

Think about planting 7 acres (3 alfalfa plus 4 wheat) at a time. Now, how many times can you repeat that within the 896 acres available? -hydnjo talk 02:58, 3 December 2008 (UTC)

So you divide? —Preceding unsigned comment added by 70.250.226.169 (talk • contribs) 03:05, 3 December 2008

For starters, yes. -hydnjo talk 03:19, 3 December 2008 (UTC)

Then what do I do?

Well each "patch" of 7 acres has some (3 acres) alfalfa and some (4 acres) wheat. After you figure out how many "patches" get planted then what do you think you should do? -hydnjo talk 04:03, 3 December 2008 (UTC)

I'm sorry man I just don't how to solve the problem. I don't know what to do next or what problem solving technique this is! How do you divide 896 acres in to a ratio of 3 alfalfa to 4 wheat? --70.250.226.169 (talk) 04:36, 3 December 2008 (UTC)

We're really not supposed to do your homework for you, but you do seem pretty desperate. Here's how you do it - the alfalfa is 3/7 of the total, and the wheat is 4/7. So you divide 896 by 7 (thats 128). Then you take 128 and multiply by 3 to get the number of alfalfa acres, and multiply 128 by 4 to get the wheat acres. And that's all. flaminglawyer^cneverforget 04:44, 3 December 2008 (UTC)

Thank You!!!

Not certain we've helped you do much. It may be that you you just don't see the relevance so you just see it as fiddling numbers around. How about if you had to divide 315 gold coins with you being entitled to 2 for every 3 a friend got? I bet you can figure out pretty exactly how many gold coins you'd be entitled to. Dmcq (talk) 11:53, 3 December 2008 (UTC)

Last two digits of a given index

How can the last two digits of any no. raised to any power be evaluated? (e.g. the last 2 digits of 3^1033 or 23^6007) I looked everywhere I could, but most of the explanations are through modular arithmetic, something I don't quite understand too well. One of my teachers told me about trying to break down the base no. into two separate nos., one of which would be a multiple of 10 (such as 23=10*2+3); evaluating the last two digits via subsequently obtained binomial expansion becomes (supposedly) easy as only final two terms of the expansion need to be taken into account (others become multiples of 100). But the remaining part still seems pretty difficult. Any ideas?--Leif edling (talk) 06:50, 3 December 2008 (UTC)

For this (the first one) specific case: 3²=9, 3⁴=9²=81, 3⁸=81²=6561=65×100+61. In further multiplications by any natural number the two digits '65' will contribute to hundreds and higher orders, so we can drop them for calculating two least significant digits: 3⁸≡61 (mod 100). Such relation is called a congruence relation in arithmetics.

Then: 3¹⁶≡61²=3721≡21 (mod 100), and so on, until 3¹⁰²⁴≡81 (mod 100).

The exponent equals 1033=1024+8+1, so finally 3¹⁰³³=3¹⁰²⁴×3⁸×3≡81×6561×3≡81×61×3=14823≡23 (mod 100)

For more information see Modular arithmetic. --CiaPan (talk) 12:23, 3 December 2008 (UTC)

As an initial go at the problem, it can help to work numerically and look for a pattern. Let's start with the number 1, and repeatedly multiply by 3, discarding all but the last two digits after each multiplication. We get:

1, 3, 9, 27, 81, 43, 29, 87, 61, 83, 49, 47, 41, 23, 69, 7, 21, 63, 89, 67, 1, 3, 9...

It should be clear that once we have gotten to 1 again the pattern will continue to repeat. Notice that this happened after precisely 20 multiplications by the number 3, that is to say, after multiplying by $3^{20}$ (in fact, it certainly must have happened within 100 multiplications, because there are only 100 two digit numbers). So what we have observed (loosely) is that multiplying by $3^{20}$ does not change the last 2 digits of a number, i.e., what we have shown is that $3^{20}\equiv 1\mod 100$, in the language of modular arithmetic.

How does this help us? Well, we are trying to calculate $3^{1033}$. We know that multiplying by 3 20 times has no effect, so clearly multiplying by 3 1020 times will have no effect 51 times, and doing nothing 51 times is the same as doing nothing. In other words, the last two digits of $3^{1020}$ is 1:

$3^{1020}=3^{20\cdot 51}=(3^{20})^{51}\equiv 1^{51}=1\mod 100$.

This is enough information to make calculating $3^{1033}$ doable. We start with the number 1, and multiply by 3 1033 times, discarding all but the last two digits each time; we see that the numbers we get repeat after every 20 multiplications, so instead of multiplying by 3 1033 times, we multiply by 3 13 times, which is small enough to do easily (it is the 14th number in our list above). In the notation of modular arithmetic, we have

$3^{1033}=3^{1020+13}=3^{1020}\cdot 3^{13}\equiv 1\cdot 3^{13}=3^{13}\equiv 23\mod 100$.

A more sophisticated version of the above approach is to use Euler's Theorem. Euler's Theorem saves us the trouble of the initial computation of the smallest power of 3 that is congruent to 1 by telling us a particular power of 3 that is guaranteed to work, namely, $\phi (100)=40$. In particular, we have

$3^{1033}=3^{1000+33}=3^{1000}\cdot 3^{33}=(3^{40})^{25}\cdot 3^{33}\equiv 1\cdot 3^{33}=3^{33}\equiv 23\mod 100$.

CiaPan demonstrated another, unrelated technique, called repeated squaring that does not use Euler's Theorem and does not require finding a power of 3 which is congruent to 1 modulo 100. Eric. 131.215.158.213 (talk) 12:56, 3 December 2008 (UTC)

Standard Error of Products

I know the Expected value of X * Y is E(x)E(y) if x and y are independent, but what about the SE(X*Y)?? 169.229.75.140 (talk) 08:02, 3 December 2008 (UTC)

See error propagation for the answer to this and more. Dragons flight (talk) 08:38, 3 December 2008 (UTC)

Basis of K(x)

The polynomials (thought as a vector space over the base field) K[x] have a basis $\{1,x,x^{2},...\}$. And what about rational functions/expressions K(x)? I'm thinking of the partial fraction decomposition, but am unable to say exactly how the basis would look like. (Would it be $x^{n}$ and $x^{k}/p(x)^{n}$ where k<n? And would they be linearly independent?) 212.87.13.77 (talk) 15:50, 3 December 2008 (UTC)

No idea, but of course the best possible partial fraction decomposition will depend on K, particularly on whether K is algebraically closed (such as C) or not (such as R). -- Jao (talk) 16:19, 3 December 2008 (UTC)

IIUIC, standard theorems on existence and uniqueness of partial fraction decomposition indeed imply that for any field K, a basis of K(x) over K can be obtained by taking all monomials x^k as well as the functions $x^{k}/p(x)^{n}$, where p is a monic irreducible polynomial, n > 0, and k < deg(p). — Emil J. 16:47, 3 December 2008 (UTC)

I had the same thought - although I think that is a basis over the quotient field of K, not over K itself (but that is probably what the questioner meant anyway). Gandalf61 (talk) 16:54, 3 December 2008 (UTC)

How is the quotient field of a field different from the field itself? — Emil J. 17:10, 3 December 2008 (UTC)

Good point ! Gandalf61 (talk) 23:26, 3 December 2008 (UTC)

Mathematical symbols like < and > but with curved lines

I tried to look up a pair of mathematical symbols I've encountered occasionally in the Maths/Stats books I read, currently "Statistical Inference under Order Restrictions" by Barlow, Bartholomew, Bremner, and Brunk, 1972, John Wiley. The symbols look like either a 'less than' <, or 'greater than' >, but the two lines of each symbol are flared like the bell of a trumpet. The authors use both the straight and curved types, clearly implying that they have different meanings. I did not find them in Wikipedia's list of mathematical symbols. Maybe someone would like to advise? These things are very hard to look up and there are no hints in the book. Thanks in anticipation86.53.57.28 (talk) 21:45, 3 December 2008 (UTC)

I'm guessing you mean $\succ$ and $\prec$? As far as I can work out, these indicate successor and predecessors, which tend to relate to either an order or a causal structure - as suggested by Forcing (recursion theory), Causal structure and Preference. It seems to just be a way of showing an ordering while reminding people that it's not a normal "greater-than/less-than" relationship like in the reals. Does that help? Confusing Manifestation(Say hi!) 22:00, 3 December 2008 (UTC)

Well, yeah, mostly. The curly-less-than is not actually standard in forcing, by the way, though I can't be certain recursion theorists don't use it that way.

However there are some more standardized meanings. For example curly-less-than (or curly-less-than-or-equal-to) is very standard in set theory and model theory for "is an elementary submodel of". --Trovatore (talk) 07:44, 4 December 2008 (UTC)

Confusing Manifestations has managed to type the symbols I am referring to. I think I get the gist. I'm going to read "less (more) than, but in a broader sense than for real numbers". That seems to fit with the sense of text in the book, e.g. "Let X be the finite set {x_1,. . . , x_k} with the simple order x_1 <(curly) . . . <(curly) x_k". So, as I understand, an experimental treatment, x_1, may be expected to give a lower response than, x_2, but the difference between x_1 and x_2 is not measurable, hence x_1 < (curly) x_2 is written rather than just plain old x_1 < x_2. Thanks for your contributions.86.53.57.28 (talk)

a*x=b (mod N)

How do we solve the analog of the linear equation in the modular integers. for example:

$45x\equiv 16{\pmod {91}}\,$

how I know x (or otherwise proof no such x exists)? PARI, Maxima (or Maple or Mathematica), Derive code would help a lot. Tahnks in advance. Mdob (talk) 22:58, 3 December 2008 (UTC)

Well, that amounts to solving $45x-91y=16$ for integers $x$ and $y$, so reading the article on Diophantine equations might help. -- Jao (talk) 23:31, 3 December 2008 (UTC)

Thank you, Jao. Mdob (talk) 23:36, 3 December 2008 (UTC)

[Linear congruence theorem] might also be of help Dmcq (talk) 00:07, 4 December 2008 (UTC)

Linear congruence theorem might help even more! --Tango (talk) 00:12, 4 December 2008 (UTC)

First solve the intermediate problem $45y\equiv 1{\pmod {91}}.$ Obviously y = −2 is a solution. Then x = 16y = −32 solves the original problem. Bo Jacoby (talk) 12:05, 4 December 2008 (UTC).