Wikipedia:Reference desk/Archives/Mathematics/2007 January 17

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January 17

triangles

Please help me solve this problem: A point P is inside an equilateral triangle.If the distances of P from the sides are 8,10,12 units, what is the length of the triangle? Thank you. —The preceding unsigned comment was added by 59.93.76.250 (talk) 02:11, 17 January 2007 (UTC).

Some hints: Let f(p) denote the sum of the distances from a point p to each of the three triangle edges, within the triangle. Then f(p) is a sum of three linear functions and is therefore itself linear. Also, you should be able to calculate the value of f(p) at each of the three triangle vertices. What does this tell you about f(p)? Now plug in the distances you are given for P. —David Eppstein 03:01, 17 January 2007 (UTC)

solution of triangles

In a triangle ABC,if cosA+2cosB+cosC=1,then the sides a,b,c are in which progression? Much help appreciated. —The preceding unsigned comment was added by 59.93.76.250 (talk) 02:19, 17 January 2007 (UTC).

Well, certainly you can exchange a and c, so either they are always equal or their relationship is indeterminate. Beyond that, consider the triangles with angles $\{1^{\circ },91^{\circ },88^{\circ }\}$ and $\{60^{\circ },103.2^{\circ },76.78^{\circ }\}$. To prove what they suggest you'll have to solve for $b(a,c)$ or so (using the sum of angles formula for a triangle), which involves non-trivial use of the trigonometric identities. --Tardis 16:52, 17 January 2007 (UTC)

If you can prove the right lower bound on cosA+cosB+cosC, one that is attained only for degenerate triangles, you should be (almost) done.  --Lambiam^Talk 18:05, 17 January 2007 (UTC)

4d6-drop-lowest probability distribution

What are the skew and kurtosis of the probability distribution obtained by rolling four six-sided dice and summing the highest three? NeonMerlin 04:19, 17 January 2007 (UTC)

I'm not sure if this is useful for you, but here's something that might help. 3-1, 4-4, 5-10, 6-21, 7-38, 8-62, 9-91, 10-122, 11-148, 12-167, 13-172, 14-160, 15-131, 16-94, 17-54, 18-21. --Brad Beattie (talk) 08:42, 17 January 2007 (UTC)

I find skewness = 0.0069 and kurtosis = −1.6398.  --Lambiam^Talk 08:47, 17 January 2007 (UTC)

Age difference

If person A was born on 17 Feb 1959, and person B was born on 7 April 1983, what is the date when person B is exactly half as old as person A? Not a homework assignment, since I'm person A and long past my homework days, (and I didn't do enough of my math homework in those days, or it would be obvious to me how to figure this out...). Thanks if you can help. —The preceding unsigned comment was added by 213.84.41.211 (talk) 09:59, 17 January 2007 (UTC).

In principle this is straightforward - you work out the number of days between A's birth date and B's birth date (i.e. A's age in days when B was born) then add these on to B's birth date to get a new date C. So at date C, B is as old as A was when B was born, and A is now twice as old as this. In practice, the calculation is tedious because you need to fiddle around with leap years. Roughly, A was 24 years and 7 weeks old when B was born, so date C will be about 7 weeks after 7 April 2007 - so sometime around the end of May 2007. To get an exact date, you could use a spreadsheet - for example, the DATE function in Excel can be used to work out the number of days between two dates. Gandalf61 11:23, 17 January 2007 (UTC)

So I make that May 26th, 2007? I love wikipedia. Thanks Gandalf.

To find nth term

e.g, If there are 4 soldiers. Out of that 3 are from Pakistan & one is you(India) .If all decides to suicide in order that is 1 after that 3 then 2.SO 4 ESCAPES. Suppose you are the 4th one the other will not know whether you died or not.

Q. If there are n no: of soldiers at what position you should stand to escape like 4 in the above example —The preceding unsigned comment was added by 82.148.97.69 (talk) 16:18, 17 January 2007 (UTC).

See our article on Josephus problem - I think this is the general name for this type of problem. Gandalf61 16:47, 17 January 2007 (UTC)

Numerically Calculating Distances in Perspective

I am rendering a checkerboard in one point perspective so that all the lines on the board going one direction converge at the vanishing point. I have only drawn the edge of the board closest to me and and line parallel to that edge that marks the first row of squares. So I know the location of the vanishing point, and the precise location and length of two consecutive lines that do not converge at the vanishing point, and nothing else. My intent is to render this board by computer, so I assume that I can measure every x and y distance upon a flat screen, but I can measure distance by no other method. How do I non-graphically find the distance from the second line away from me that I have drawn to the third line away from me in terms of vertical distance on a flat screen? Please let me know if this question doesn't make sense, and thank you for your attention. --Amanaplanacanalpanama 00:44, 18 January 2007 (UTC)

Depends on how far away your eye is. As you probably know from photography, the appearance of the scene will be different depending on whether it is observed from far away with a telephoto or from a shorter distance with a wide-angle lens. Imagine that you are holding a transparent screen a distance L from your eye, with the screen perpendicular to the line from your eye to the vanishing point, and that you are creating the image by tracing on this screen what you see through it. Then, if we use coordinates {x,y,z} with your eye as the origin, and the z direction being towards the vanishing point, everything becomes simple: An object at position {x,y,z} gets depicted at the point {xs,ys}=(L/z){x,y} on the screen. Just sketch this situation seen from the side, and some light rays, and this relation will become obvious. --mglg_(talk) 01:14, 18 January 2007 (UTC)

Thank you. That was imensly helpful. --Amanaplanacanalpanama

Like the factorial function, but with addition

Hi, we all know that...

5! = 5 * 4 * 3 * 2 * 1 = 20

But is there a function that does the sum instead? So...

5? = 5 + 4 + 3 + 2 + 1 = 15

Thanks -Quasipalm 01:06, 18 January 2007 (UTC)

See triangle number. --mglg_(talk) 01:15, 18 January 2007 (UTC)

Summarizing "triangle number" we can find that it simply boils down to a formula, (n(n+1))/2, with n being the number of numbers. For example, you have 5, so 5(6)/2, which is 30/2, or 15. Alternatively, for Series, known as "Arithemtic Series," you can use Summation, with the Greek letter Sigma. A GREAT article on this is found at : summation. I hope I helped! Feel free to ask more questions if you need more help. Root2 01:20, 18 January 2007 (UTC)

Awesome awesome awesome, thanks all. See, I'm building this trilateral pyramid made out of watter bottle tops on my desk, and I couldn't figure out for the life of me the formula for the volume of the pyramid... I even started making a graph...

1 |                  01                  = 1
2 |                03 +01                = 4
3 |              06 +03 +01              = 10
4 |            10 +06 +03 +01            = 20
5 |          15 +10 +06 +03 +01          = 35
6 |        21 +15 +10 +06 +03 +01        = 56
7 |      28 +21 +15 +10 +06 +03 +01      = 63
8 |    36 +28 +21 +15 +10 +06 +03 +01    = 99
9 |  45 +36 +28 +21 +15 +10 +06 +03 +01  = 144
10|55 +45 +36 +28 +21 +15 +10 +06 +03 +01= 199

Anyway, these articles totally help! I may have it shortly... -Quasipalm 01:25, 18 January 2007 (UTC)

Ok, so it's clearly $\sum _{i=1}^{n}{\frac {n(n+1)}{2}}$. Thanks again all. I put up a java program that'll calculate triangle numbers here if any is interested. -Quasipalm 02:11, 18 January 2007 (UTC)

Oh, it's 3D. Then what you get is a tetrahedral number. If you get into other geomtries after a few more bottles of water, you may want to check out the other kinds of pyramidal numbers. --mglg_(talk) 02:16, 18 January 2007 (UTC)