Question 1148232
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Let "x" be the square side length, and "y" be the radius of the circle.

Then

    4x + {{{2*pi*y}}} = 16 inches                               (1)  (perimeter)

    f(x,y) = x^2 + {{{pi*y^2}}}   is the function to minimize   (2)  (area)


In other words, you should minimize (2) under the constraint (1).


From (1), express x = {{{(16-2*pi*y)/4}}} = {{{4 - 0.5*pi*y}}}  and substitute it into the function (2).

You will find then the function to minimize in the form

    g(y) = {{{(4-0.5*pi*y)^2}}} + {{{pi*y^2}}} = 

         = {{{16 - 4*pi*y + 0.25*pi^2*y^2}}} + {{{pi*y^2}}} = {{{16 - 4*pi*y + 1.25*pi^2*y^2}}}.


This quadratic function of "y" has the minimum at  y = " {{{-b/(2a)}}} " = {{{(4*pi)/(2*1.25*pi^2)}}} = {{{3.2/pi}}}.    


<U>ANSWER</U>.  The values that provide the minimum of the total area are

         y= {{{3.2/pi}}} inches (the circle radius)  and  

         x = {{{(1/4)*(16 - 2*pi*y)}}} = {{{(1/4)*(16 - 2*pi*(3.2/pi))}}} = {{{(1/4)*(16-6.4)}}} = {{{(1/4)*9.6)}}} = 2.4 inches (the side of the square).
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Solved.