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Calculus/Applied Max/Min Problem

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Expert: Alon Mandes - 11/24/2009


Question
I have started this question but am not sure if I am doing it correctly. Your help will be greatly appreciated. :)



The sum of twice one number and five times a second number is to be 70. What numbers should be selected so that the product of the numbers is as large as possible?



The solution to this problem is 7 and 17.5, but I need to be able to show my work step by step.



This is what I have so far:



70 = 2x + 3y

14 = (2/5)x + y

35 = x + y

y = 35 - x

y' = -1

CN = 0



Is this correct? and what is my next move?



Thank you for your attention to this problem.

Answer
2x+5y=70  --> x=(70-5y)/2
we want to maximize x*y :
x*y=(70-5y)y/2=(70y/2)-5y^2/2=35y-2.5y^2
To find maximum we derive the expression & set it equals zero :
(35y-2.5y^2)'=35-5y
We solve the equation : 35-5y=0 --> y=7
Now we want to find x :
x=(70-5y)/2=(70-5*7)/2=(70-35)/2=35/2=17.5

Alon.

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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M.A in Mathematics & Bs.c in Electronics.

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