Wikipedia:Reference desk/Archives/Mathematics/2007 May 14

= May 14 =

Equation
How can you find this equation where the vertex is (4,6) and the x-intercepts are (8,0) and (0,0)? Please answer the question. thank you. —The preceding unsigned comment was added by 76.64.53.59 (talk) 01:22, 14 May 2007 (UTC).
 * We don't do homework, but your question is very ambiguous. There are an infinite number of equations that can accept the information you have given.  x42bn6 Talk Mess  01:59, 14 May 2007 (UTC)


 * I think it's safe to assume you're talking about a Parabola. The article is very informative. At least, bits of it are. Given the vertex and the x-intercepts, there are several ways to work out the equation in some form or other. One makes use of the base equation y = x^2 and adjusts from there. It starts out with vertex at (0,0), opening up. To move the graph horizontally and vertically, we introduce the vertex (h,k), making (y-k) = (x-h)^2. To stretch the graph out to the right shape, and flip it over if necessary, we introduce a constant I'll call b, making (y-k) = b(x-h)^2. To find b, plug in h, k, x, and y (any x and y will do, like one of the x-intercepts) and solve for b. Black Carrot 02:45, 14 May 2007 (UTC)


 * Perhaps this is the same person who asked about minimum value, above. If so, then a parabola is expected. But the questioner must respect the rules of this page (stated at the top), which explicitly prohibit homework solutions. What we can do is offer guidance, as Black Carrot has done. We can also confirm reasoning.
 * Besides, the way I personally would fit a parabola through three points is likely not appropriate as a homework answer. I might fit a straight line through (0,0) and (4,6), fit another through (4,6) and (8,0), then blend them to produce a parabola, in the manner of Lagrange interpolation.
 * $$\begin{align}

p_{0,4}(x) &= \frac{4-x}{4-0} 0 + \frac{x-0}{4-0} 6, \\ p_{4,8}(x) &= \frac{8-x}{8-4} 6 + \frac{x-4}{8-4} 0, \\ p_{0,8}(x) &= \frac{8-x}{8-0} p_{0,4}(x) + \frac{x-0}{8-0} p_{4,8}(x). \end{align}$$
 * But if you hand this in for homework, you will have learned nothing, the teacher will immediately suspect cheating, you will be asked to explain the answer, and you will fail. And besides, it takes no advantage of the special nature of the three points given. The advantage for me is that I can extend this method to fit a polynomial of degree n−1 through n data points, so it's a simple and handy method to remember. --KSmrqT 03:31, 14 May 2007 (UTC)
 * When he said, "vertex", I thought he might have meant something like an absolute-value graph (y=-|x+1|?). x42bn6 Talk Mess  14:52, 14 May 2007 (UTC)


 * A good point. My analysis would extend to that case nicely, though I'm afraid KSmrq's might not. Black Carrot 05:25, 15 May 2007 (UTC)
 * No worries; omit the blending step and my algorithm delivers a piecewise linear fit. A slight variation will yield B-spline curves instead of interpolating curves (as in the de Boor algorithm), and a combination of the two can deliver a family of different types of curves, including piecewise cubic interpolants with first derivative continuity. --KSmrqT 10:07, 15 May 2007 (UTC)

Additional MathsGCSE question on Vectors
(Throughout thisquestion i and j denote unit vectors parallel to standard set of x-y axis)

A bodyof mass 2kg is in equilibrium under theaction of three coplanar forces P,Q and R where

P=(3xi-2yj)N       Q=(2yi+4xj)N       R=(-11i+4j)N

(i)Calculate the values of x and y

Cone-Pine1991 14:14, 14 May 2007 (UTC)
 * Do your own homework. However, a hint: P+Q+R=0, allowing you to find the values of x and y.  x42bn6 Talk Mess  14:50, 14 May 2007 (UTC)

Proof of Cone and Sphere Surface Areas
Hello. Can anybody please tell me the proof of the right cone's and the sphere's surface area? I am only in middle school, so please keep the explanation simplified. Thanks. --Mayfare 18:01, 14 May 2007 (UTC)


 * Are you familiar with calculus? It could be tough to explain otherwise. -- Meni Rosenfeld (talk) 18:58, 14 May 2007 (UTC)
 * For the cone, I suggest you first take a look at Cone (geometry). If you're wondering why the lateral area is $$\pi r s$$, the intuitive explanation is: You start with a circle of radius r (and perimeter $$2 \pi r$$), and you take it along a distance of s while shrinking it to a point. The average perimeter is $$\pi r$$, and you multiply the perimeter with the length to obtain the area. -- Meni Rosenfeld (talk) 19:10, 14 May 2007 (UTC)


 * For the cone there is an elementary derivation, although it uses a bit of handwaving. Let r be the radius of the disk making up the base, and s = √(r2+h2) the slant height (the distance from any point on the circle that is the boundary of the base to the apex of the cone), where h is the standard height. The formula A = πr2 + πrs gives the area as the sum of two terms. The first, πr2, is simply the area of the base disk, which is flat. The second, πrs, then must be the area of the surface formed by the lines to the apex, which is curved in 3D space. Its intrinsic curvature, however, is 0: we can cut this surface open along a line from any point on the circle to the apex, and then flatten it. (Here is where the handwaving comes in; I have not actually given a proof that the surface is flattenable.) The result is a circular sector, like a pizza slice. The disk of which this is a segment has radius s. The arc length of the curved boundary is the length of the original circle, 2πr. The relationship between the arc length a and the angle θ of the sector is a = sθ, or θ = a/s. The area of the sector is 1/2θs2. Using θ = a/s and a = 2πr, we have θ = 2πr/s, so the area is 1/2θs2 == 1/2(2πr/s)s2 = πrs. Q.E.D. --Lambiam Talk  22:40, 14 May 2007 (UTC)
 * And for that you already need that the area of a circle is πr2, which I don't think can be proven elementarily without handwaving either. Anyway, I think someone should point out that without calculus (or something like it), it's not clear how we can even define surface areas of curved objects, let alone actually calculate them! One can assign area to flat polygons easily enough by cutting them up into triangles, but how do you extend this to circles, let alone non-flat surfaces? Algebraist 00:05, 15 May 2007 (UTC)


 * (edit conflicted) Archimedes famously found the surface area of a sphere using only the area formulae for the cone and for the conic frustum. He inscribes a solid composed of frustums (and cones on the two ends) in a sphere. Projected into two dimensions, the solid looks like a an n-gon inscribed in a circle. He finds the surface area of the solid and then lets n go to infinity. The details of the proof are involved, but I think the intuition is clear. As a historical note, Archimedes did not have access to limits the way we do, so his proof used a proof by contradiction: he showed the s.a. of a sphere cannot be strictly more or strictly less than "four times its greatest circle." Archimedes was so proud of his result that according to legend, he had asked for a picture of a sphere inscribed in a cylinder to be carved opon his tombstone. See if you can find the connection. (I took this info from Dunham's The Mathematical Universe) nadav 00:18, 15 May 2007 (UTC)
 * This is what I meant by something like calculus above. The classical greeks (who were far more rigorous in their mathematics anyone else until at least the mid 19th century) would not have been happy with any handwaving, and developed a cunning method for this sort of problem. However, the method (being based as you say on contradiction) could only be used to prove results and not find them. This is especially telling in the first use of the method, when Eudoxus proved that the area of a circle is proportional to the area of the square on its diameter, a result that is blindingly obvious as long as circles have area at all. Thus the purpose of the technique was to put results on a sound footing, not to discover them. For this Archimedes used an entirely different method, with more handwaving than a busload of queens. Algebraist 00:37, 15 May 2007 (UTC)
 * Strange image; I wonder if this is what you mean? Or perhaps a bus in Queens, NY?
 * As for the mathematics, see area of a disk. --KSmrqT 02:13, 15 May 2007 (UTC)
 * Lol. I assume he meant the first one. I love that metaphor. Did you come up with it yourself, Alg? I reserve the right to steal it since it's published under GFDL (but I'll give you credit). nadav 02:42, 15 May 2007 (UTC)
 * KSmrq, your first image fits the waving part, but this one is what first came to mind for me... —David Eppstein 03:47, 15 May 2007 (UTC)
 * Having seen both this (recommended) and this (an acquired taste I never acquired), I had two obstacles: (1) where to find the bus (well done!), and (2) although anyone with the chutzpah to be on that bus would hardly be offended, I didn't want to feed the dark side (intolerance) of humanity. --KSmrqT 10:55, 15 May 2007 (UTC)
 * I was indeed thinking of the main purpose of existence of Her Britannic Majesty, Elizabeth the Second, by the Grace of God of the United Kingdom of Great Britain and Northern Ireland and of Her other Realms and Territories Queen, Head of the Commonwealth, Defender of the Faith, Lord of Mann, Duke of Normandy, Sovereign of the Most Noble Order of the Garter, Fellow of the Royal Society, freewoman of the City of Philadelphia etc etc, but am happy for its meaning to be ruthlessly reinterpreted per the GFDL. Algebraist 15:05, 15 May 2007 (UTC)


 * I have elsewhere described KSmrq's second example as "absolutely the vilest film ever made". Fortunately, it's not representative of anything but itself.  JackofOz 02:40, 16 May 2007 (UTC)