Calculus/optimizing

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Expert: Alon Mandes - 11/23/2008


Question
Hi... Im hoping you would help with problem i tried so many times to do but im glad i did the rest of this take home quiz..

A company wants to spend 4$ per cylindrical can in their manufacturing plant. the material for the top and bottom of the can costs $1.50 per square inch and the material for the sides costs 0.75$ per square what dimesions will maximize the volume of the can?.

thank you so much for the effort you guys put to help us ...

Answer
Let's review some of the facts :
1. The area of the two base is : B=2*π*r^2
2. The area of the side is : S=2*π*r*h
3. The total can material is : B+S=2*π*r^2+2*π*r*h.
We claim that : (1.5)B $ + (0.75)S $ = 4$
That means : (1.5)(2*π*r^2) $ + (0.75)(2*π*r*h) $ = 4$ .
3πr^2+(3/2)πrh=4 -> h=[4-3πr^2]/[(3/2)πr].
The volume of the can is : V=πhr^2=πr^2[4-3πr^2]/[1.5πr].
V=r[4-3πr^2]/1.5=(8/3)r-2πr^2. Now we want to maximize the volume.
So we calculate V'=0 -> V'=(8/3)-4πr. -> V'=0 -> r=(2/3π) ->
h=[4-3π(4/9π)]/[(3/2)π(2/3π)]=2.66

Alon.  

Calculus

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Alon Mandes

Expertise

Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

Experience

1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

Organizations
Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

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