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Question
A closed cylindrical canister or volume 8(pi) cubit feet is to be made. It must stand between two partitions 8 feet apart in a room with an 8-foot ceiling. If the cost of the top and bottom is $1 per square foot and the cost of the latter surface area is $2 per square foot, find the dimensions of the least expensive canister.
The volume of the canister V=πr²h & it's equals 8π . Therefore r²h=8 --> h=8/r².
The total area of the canister consists of : base+bottom+surface. The total coast will be:
S=base*1$+bottom*1$+surface*2$
S=πr²$+πr²$+2πrh*2$=2πr²+4πrh ($)
We substitute h as function of r , we get :
S=2πr²+4πr(8/r²)
S=2πr²+32π/r
To find minimum, we derive :
S'=4πr-32π/r²
We solve S'=0 :
4πr-32π/r²=0
r=8/r²
r³=8
r=2.
Let's find h: h=8/r²=8/4=2 .
The dimensions are : r=2 & h=4 .
Alon.
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Answers by Expert:
Alon Mandes
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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 .
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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".
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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.
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M.A in Mathematics & Bs.c in Electronics.
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