Advanced Math/Radius

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Question
Bonjour,

This is not homework.

We want to calculate the surface area of a sphere as it decreases down towards the center of the sphere.  Multiplying surface area by the radius would given an incorrect result as that would not encompass the surface area getting smaller.

Multiplying the full surface area by what  "percent of the radius"  would give a correct value ?

Thanks so much,
Anne

Answer
The surface area of a sphere is given by the formula

As = 4(pi)(radius)^2

By "decreasing down toward the center of the sphere" I believe you mean "what is the surface area of the sphere as the radius goes to 0". In this case you could just plug in the desired value of radius = r to get your answer.

It seems like you want to start with a sphere of a given radius, say R, and multiply it by a factor involving the new radius to get the area as the radius goes to zero. So, lets start with

AR = 4(pi)R^2   = area of sphere of radius R

and then try to end up with

Ar = 4(pi)r^2.

This can be done by

Ar = AR(r/R)^2

so you are multiplying by the ratio of the radii. This could be interpreted as a "percent of the radius".

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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