If you had a piece of wire 16m long and is cut into two pieces. One piece is bent into a square and the other is bent intoa circle.Express the total area A of the square and circle as a function of X, the length of the sides of the suare.How shoud the wire be cut so that the total are enclosed is a minimum, and what is the total area enclosed?

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Lina Watson · Accepted Answer

so we have 16m wire, which we will cut into 2 pieces. Since we are using x as the perimeter of the square. we will have perimeter ofsquare = x and circumference of circle = 16-xequations for circles  π      r          2        =  A  r  e  a  2  π  r  =  C  i  r  c  u  m  f  e  r  e  n  c  e  r  =  (  16  −  x  )  ÷  2  πEquations for SquaresArea=l*l Perimeter = 4lx=4LL= x/4  A  r  e  a  =  (  x      /    4  )  (  x      /    4  )  =  (      x    2    )      /    16Therefore total Area = Area of Square + Area of CircleTotal Area   =            x              2              16    +                    x                  2                    −      32      x      +      256              4      π      From the equation above, we should combine them (since this is algebra and not calc, other wise you could find the first and second derivatives).Total Area  =            (      π      +      1      )              x                  2                    −      32      x      +      256              16      π      The min/max(x) will occur at -b/2a  −  b      /    2  a  =  32      /    (  2  π  +  2  )Then you just plug in this value for x. This minimizes the area and provides you will what the total area when minimized.

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