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Subject: "equations"     Previous Topic | Next Topic
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Conferences The CTK Exchange High school Topic #268
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^2
dx SA = 8 pi r
Circumference = 2 pi r

never thought of it this way and I can't find why.

plz share some insight, thanx


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jada_vis@msn
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)


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

DSCW = 2pRDr

the same as the surface of the sphere slice wall

DSS = 2pR cosqDr/cosq

Since the total area of the cylinder wall is

S DSCW = 2pRS Dr = 2pR2R = 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 - 4pR2 - 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 = 2pRDr

If you have 4 circles of radius R and do the same thing, the total area increase is

4DA = 42pRDr = 8pRDr

and 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

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