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 Subject: "equations" Previous Topic | Next Topic
 Conferences The CTK Exchange High school Topic #268 Printer-friendly copy     Email this topic to a friend Reading Topic #268
James
guest
Oct-02-03, 11:53 PM (EST)

"equations"

 I came across a weird question today. It was "explain why the derivative of the surface area equation of a sphere is 4 times the circumference of the sphere?"SA = 4 pi r^2dx SA = 8 pi rCircumference = 2 pi rnever thought of it this way and I can't find why.plz share some insight, thanx

guest
Oct-04-03, 11:49 AM (EST)

1. "RE: equations"
In response to message #0

 Posted in this site: Archimedes ....S= (4pi)r^2 {see: Language of Math)

Member since Jun-22-03
Oct-06-03, 03:23 PM (EST)

2. "RE: equations"
In response to message #0

 LAST EDITED ON Oct-06-03 AT 04:04 PM (EST) The first sketch shows how, most likely, Archimedes figured out the surface of a sphere: Put a sphere of radius R in a cylinder of the same radius R and height 2R. Cutting both figures into thin horizontal slices of height Dr, the surface of the cylinder slice wall isDSCW = 2pR·Drthe same as the surface of the sphere slice wallDSS = 2pR cosq·Dr/cosqSince the total area of the cylinder wall is S DSCW = 2pR·S Dr = 2pR·2R = 4pR2 so is the surface of the sphere. Then he stated that the surface of a sphere is 4 times the area of its greatest circle - 4´pR2 - see Mathematics as a Language. Now, if you have a circle of radius R and you increase the radius by Dr, the area increase is (see the second sketch) DA = 2pR·DrIf you have 4 circles of radius R and do the same thing, the total area increase is4·DA = 4·2pR·Dr = 8pR·Drand according to Archimedes' statement, this is the same surface increase as if you have a sphere of radius R and increase its radius by Dr.Regards, Vladimir