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Calculus/calculus - optimization

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Expert: Alon Mandes - 4/11/2010


Question
a cylinder, open at one end and having a surface area of 243п cm^2 is to be made from sheet metal. if the volume is to be a maximum, find the radius and also the volume.

how do i solve this!?

Answer
The surface area of a cylinder is 2πr²+2πrh . In our case it will be S=πr²+2πrh .
We know that : S=243π . Thus, πr²+2πrh=243π --> h=(243-r²)/(2r)=[243/2r]-[r/2] .
The volume of the cylinder is : V=πr²h . Substituting the relation between r & h :
V(r)=πr²([243/2r]-[r/2])=[243πr/2]-[πr³/2] . To find maximum we derive V(r) & solve V'(r)=0 :
V'(r)=[243π/2]-[3πr²/2] .
V'(r)=0 --> [243π/2]=[3πr²/2] --> 243=3r² --> r=9 cm .
The maximum volume will be Vmax=π*(9²)*[243/(2*9)]-[9/2]=729π

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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