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Optimization Problems

A piece of plexiglass is in the shape of a semicircle with radius 2 m. Determine the dimensions of the rectangle with the greatest area that can be cut from the piece of plexiglass. 

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Howard L. | Experienced Math Tutor Specializing in Test Prep & Grade Up SkillsExperienced Math Tutor Specializing in T...
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The circle with radius 2m :  x^2 + y^2 = 4
The semicircle:  y=sqrt(4 - x^2)
The Area: A = 2xy = 2x(sqrt(4 - x^2))
Find A', simplify, and set 0:  
        x^2 - 2 = 0, and x = sqrt(2), y = sqrt(2)
Take A'' < 0  to verify maximum area.
 
Therefore, base length = 2x = 2 sqrt(2)  meter                    
                width = y = sqrt(2) meter
                area = 4  square meter 
 
Cut sqrt(2) away from the center both +, -  x direction, and + y direction!