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Expert: Alon Mandes - 1/19/2009


Question
Given an open-top box with a rectangular base x by y units and rectangular sides with depth z units and holding the depth constant, show that the maximum volume of the box for a given amount of material occurs at x=y.

Answer
We know that the volume V=xyz. We also know that the surface
S=2xy+2xz+2yz=2xy+2z(x+y). The amount of material is constant. That
means Constant=2xy+2z(x+y) Or z=(C-xy)/xy where C is another constant. Now, lets calculate partial derivatives :
∂z/∂x=[ -(C-xy)-y(C-xy) ] / (x+y)² = [-C+xy-yC+xy²) ] / (x+y)²
∂z/∂y=[ -(C-xy)-x(C-xy) ] / (x+y)² = [-C+xy-xC+yx²) ] / (x+y)²
Maximum occurs when ∂z/∂x=∂z/∂y . Hence, xy²=yx² -> y=x.

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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