Optimization Problems

Question

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. 

Answer 1

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!