Wikipedia:Reference desk/Archives/Mathematics/2009 December 15

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December 15

Isoperimetric yard problem

"The management of Home Depot has 1600 feet of fencing to fence in a rectangular storage yard using the building as one side of the yard. If the fencing is used for the remaining 3 sides, find the area of the largest possible yard."

Okay, I'm having a bit of difficulty with this problem. I have attempted to set up a primary equation and a secondary equation. Primary: A=xy Secondary: 2x+y=1600

How am I doing so far?

Solving for y yields y=1600/2x and substituting the answer into the primary equation gives A=x(1600/2x) which simplifies to A=1600/x

I calculated A'(x)=-1600/x^2 ....but now what do I do? I'm a little lost. Thanks for the help! —Preceding unsigned comment added by 98.108.33.148 (talk) 00:35, 15 December 2009 (UTC)

2x+y=1600 does not solve to y=1600/2x. Try plugging some numbers in. Where'd you get a division from?

Later on, x(1600/2x) does not simplify to 1600/x (unless x happens to be 2). –Henning Makholm (talk) 00:48, 15 December 2009 (UTC)

Oh my gosh! Thanks so much, I misread my own equation! I got the maximum area to be 320,000 ft squared. Thanks again Henning I appreciate it :) —Preceding unsigned comment added by 98.108.33.148 (talk) 01:13, 15 December 2009 (UTC)

That's OK as long as the side of the building is 800 feet long or more. If the side is y (<=800), the fenced area will be y(1600-y)/2 (<=320,000)——86.164.72.171 (talk) 15:49, 15 December 2009 (UTC)

• But surely the length of the building is fixed. If the building has length b and it is to make a side of the yard then the side of the fence which is perpendicular to the side of the building will have length ½(1600 − b). So the area of the yard will be ½(1600 − b)b = 800b − ½b². This is the maximum area because then lengths of two sides of the rectangular yard are fixed by the length of the building. The lengths of the other two sides come by using up all of the possible fencing. If the size of the building is allowed to change, say we are planning to build the depot, then we have a function of b. Let ƒ(b) = 800b − ½b². This has an extremum when dƒ/db = 0, i.e. when b = 800. This extremum is a maximum if d²ƒ/db² < 0 when b = 800. Well, d²ƒ/db² = −1 < 0 so b = 800 is a maximum. In fact, since ƒ is quadratic it is a global maximum. In this case the yard is 400-by-800 giving an area of 320,000 feet². ~~ Dr Dec (Talk) ~~ 16:17, 15 December 2009 (UTC)

• A more interesting problem might be to have a wall of the depot contained as the middle of one of the sides of the rectangular yard (instead of the wall of the depot making up a whole side of the yard). In that case, with a total of 1,600 feet of fencing, I get the maximum yard area to be (¼b + 400)². Can you see how? ~~ Dr Dec (Talk) ~~ 16:33, 15 December 2009 (UTC)

rotated figure

A triangle is rotated in plane through an angle (clock wise or anti clock wise ), point P as centre of rotation.? if only object triangle and image triangle is given .How can we find out centre of rotation only by observation.( I know the method of point of intersection of perpendicular bisectors ).--True path finder (talk) 02:05, 15 December 2009 (UTC)mks

We can't, unless P is actually on the triangle, in which case it is the invariant point and will stand out quite clearly (you just need to look at the points where the two triangles meet and you should be able to tell which of those is the centre). --Tango (talk) 02:42, 15 December 2009 (UTC)

Tango, to say that it can't be done is to say that more than one rotation would give the same result. Can you exhibit and example of that? Michael Hardy (talk) 04:45, 15 December 2009 (UTC)

The OP said "only by observation". Unless P is a point of intersection of existing lines there is no way to identify it without drawing new lines. If you are allowed to draw new lines you can do it very easily (the OP mentions one method), but that isn't the question being asked. --Tango (talk) 16:26, 15 December 2009 (UTC)

If the rotation angle is $\pi$, there are two such axis. Generally, I believe it is but one. See this for a discussion of Chasles' theorem, and furthermore a way to find rotation axes - what I recommend is taking the rotation about some point on the triangle as an axis of initial rotation, followed by some translation of that same point to bring the triangle to its image. Then, you can work out an axis and angle of rotation.--Leon (talk) 06:07, 15 December 2009 (UTC)

Further: if the triangle is equilateral, you may have some problems. Owing to the symmetry of an equilateral triangle, a rotation of $2\pi /3$ may be indistinguishable from one of $4\pi /3$. So solutions won't be unique in any case.--Leon (talk) 06:13, 15 December 2009 (UTC)

Let the triangle be ABC and A'B'C'. The point P must be equidistant from A and A' and also equidistant from B and B', so P must lie on the intersection of the perpendicular bisectors of AA' and BB' and as a corollary, the perpendicular bisector of CC' falls on the same point. I assume that's what the OP means by the method of perpendicular bisectors but it seems like the easiest way to do it. The case of an equilateral triangle may give problems if you don't keep the labels on the vertexes, but with a labeled triangle there's no problem.--RDBury (talk) 07:35, 15 December 2009 (UTC)

Be careful of the case when P lies on the (extended) line AB; in this case the perpendicular bisectors of AA' and BB' will be coincident, and you need the perpendicular bisector of CC' to pick out a particular point on this line. Also, I don't follow Leon's point about there being more than one candidate for P when the angle of rotation is 180 degrees - we are talking here about plane geometry, not spherical geometry, aren't we ? Gandalf61 (talk) 09:56, 15 December 2009 (UTC)

I'd search for this spot marked 'P' that it was rotated around. Sorry couldn't resist that :) Dmcq (talk) 18:44, 16 December 2009 (UTC)

Question in computing probabilities

I think this is a pretty simple problem in probability, but I can't figure it out:

A handwriting recognition software uses n heuristic algorithms to test a user defined input. The software is designed to return 26 probabilities indicating to what extent the software thinks that the user defined input is each of the 26 letters in the alphabet (these 26 values will sum to 1). After the software is run, each of the n heuristics return 26 values, each value indicating the probability that the unknown letter is each of the 26 output possibilities (these values obviously sum to 1 as well). Therefore, after processing a piece of user defined input, the software returns an array of n probabilistic indicators for each of the 26 possible outputs. In other words, the software returns a double array of size 26*n, populated with probabilities between 0-1. How are these probabilities in this double array computed to arrive at final probabilities that the user defined input is each of the 26 possible letters?

Here is a concrete example of what I am describing: Lets say the algorithm uses five heuristics to asses user defined input. The user shows an input to the algorithm. The first heuristic may return a .1 probability that the input is an “A,” a .7 probability that input is a “B,” etc for all 26 letters in the alphabet. This happens for each one of the 5 heuristics. How is this data computed to give one final probability number for each of the letters in the alphabet? —Preceding unsigned comment added by 68.175.97.21 (talk) 04:11, 15 December 2009 (UTC)

It seems to me that you don't have enough information to solve the problem because there is nothing about accuracy of each heuristic. For example, with two heuristics, the first one might be really good at figuring out the letters and the second one might be just spitting out random numbers, in which case you should be throwing away the input from the second heuristic and using the first. Or it could be the other way around in which case just use the second heuristic. From what you've given in the problem you don't know that these extremes can't happen so it's impossible to give a solution. You need to have in addition some kind of track record for each heuristic of how accurate it is and which letters it does best at. Once you have that info then a solution might be possible, but it sounds like the kind of thing you find in papers on neural networks etc.--RDBury (talk) 07:59, 15 December 2009 (UTC)

The answers above at #weighted mean and errors may be of use, In actual practice as the previous responder has said a neural network would probably be used, and it may cycle around trying to get to a better guess. There is a related problem in error detection and removal where nowadays they quite often use a few different algorithms and decide amongst them rather than as previously have something nice and mathematically worked out. By the way with handwriting they can mistake two letters as one or split a letter in half by mistake as well never mind all the funny signs people use. Optical character recognition doesn't say very much about how it is done. If you have a look at Machine learning which is what underlies much of it you'll see how complicated it has all become. Bayesian inference is what is at the bottom of most of it. Dmcq (talk) 11:22, 15 December 2009 (UTC)

Logarithms and Conic Sections.

I am entering a college algebra course next semester and I feel that my previous algebra course may not have provided me with all the skills needed to succeed. My instructor pretty much skipped or skimmed over the Logarithm and Conic section portion of the book in an effort to "help" the struggling students. What are these used for? How are they used? Are they going to be an important part of my next course? Math is not my strongest subject, but I want to be prepared as possible, as I am required to take trig and a calculus course. Thank you. —Preceding unsigned comment added by 161.165.196.84 (talk) 12:07, 15 December 2009 (UTC)

Logarithms are extremely important - it is likely you will have a lot of trouble if you don't familiarize yourself with them. Our article on the subject may be a good start.

Conic sections are not that important, you may want to hold off reading up on them until you know they are required. -- Meni Rosenfeld (talk) 14:46, 15 December 2009 (UTC)

I agree, logarithms are essential for a large portion of mathematics. I don't remember using Conic sections in anything other than a topic on Conic sections. They are a very interesting part of geometry and they do have applications, but not ones that tend to come up in undergraduate mathematics sources. --Tango (talk) 18:27, 15 December 2009 (UTC)

Normal variables, independence, covariance

For two normal variables, are independence and covariance being 0 equivalent? I think I remember this but I can't find it in my book. Thanks. StatisticsMan (talk) 22:50, 15 December 2009 (UTC)

No. Two variables whose joint distribution is normal are independent iff uncorrelated. See Normally distributed and uncorrelated does not imply independent. Algebraist 23:22, 15 December 2009 (UTC)