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Question
Can you assist please?

A manufacturer wishes to produce the most economical packaging for detergent. The containers need to hold 25litres and must be cylindrical in shape.
What dimensions will ensure that the least amount of plastic is used to create the containers?
(Note: the volume and surface area of cylinders are V = πr^2h and S = 2πr^2 + 2πrh respectively)

Answer
The volume must be 25

π r^2 h = 25

h = 25/πr^2

Surface area

S = 2πr^2 + 2πrh


Substitute 25/πr^2 for h

S =  2πr^2 + 50/r

S' = 4πr - 50/r^2

0 = 4πr - 50/r^2

0 = 4πr^3 - 50

r = (25/2π)^1/3

The first derivative test shows that this value for r gives the minimum surface area

Since h = 25/πr^2

Substitute r = (25/2π)^1/3

and get h = (100/π)^1/3


Since a liter is one cubic decimeter , the units for r and h are decimeters

Calculus

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I can answer questions from the standard four semester Calculus sequence. I am not prepared for questions on Tensor Calculus. Everything else is welcome. Derivatives, partial derivatives, ordinary differential equations, single and multiple integrals, change of variable, vector integration (Green`s Theorem, Stokes, and Gauss) and applications.

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Ph.D. in Mathematics and many years teaching Calculus at state universities.

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B.S. , M.S. , Ph.D.

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