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Calculus/Minimizing Cost

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Expert: Alon Mandes - 3/18/2009


Question
Hello Chanda,
I am having a time with this question, please help.  A design for a bus stop shelter has 2 square sides, a back and a top.  The materials cost $7 per square foot for the top and $3 per square foot for the sides and back.  If the volume is to be 1200 cubic feet, find the dimensions that minimize the cost.

Answer
The total material required is : A=2(Side)+1(Top)+1(Base).
If we set x as the length of each side, & y is the length of the
top then :
Coast=3$*2x²+7$*xy+3$xy=6$*x²+10$xy .
Now, we know also that the volume is 1200, thus :
x²y=1200 -> y=1200/x² .
Now the coast function becomes :
Coast=6x²+10x*(1200/x²)=12x²+10/x . Let's derive :
Coast'=12x-10/x² . To find minimum, we set Coast'=0. Therefore :
12x-10/x²=0 -> 12x=10/x² -> 12x³=10 -> x³=5/6 -> x=0.941 foot.
Hence, y=1200/0.88=1355 feet.

Calculus

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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.

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

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