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Expert: Alon Mandes - 2/1/2010


Question
A window consisting of a rectangle topped by a semi circle is to have an outer perimeter P. Find the radius of the semi circle if the area of the window is to be a maximum.

Answer

drawing
I attached an image or a drawing to illustrate the problem .
P=2x+2y+2πr . Since r=y/2 , then P=2x+2y+πy --> y=(P-2x)/(2+π) .
The area is S=x*y+πr²=[(Px-2x²)/(2+π)]+πy²/4 = [(Px-2x²)/(2+π)]+(π/4)[(P-2x)/(2+π)]²
Let's derive S %26 solve S'=0 :
S'(x)=[(P-4x)/(2+π)]+2(π/4)[(P-2x)/(2+π)][(-2)/(2+π)]
S'=0 : P-4x-π(P-2x)=0 --> P-4x-πP-2πx=0 --> P(1-π)-x(4+2π)=0 --> x=(1-π)/(4+2π) .
%26 from here you can calculate r .

Alon.

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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