Question 156612
One of the pieces is x mm long and the other is 350 - x mm.


Let's take the first one and make a circle.  That means the circumference of the circle is x, but we know that the circumference of a circle is {{{2*pi*r}}}, so the radius of the circle must be {{{r=x/(2*pi)}}}.  The area of a circle is {{{pi*r^2}}}, so the area of this circle must be {{{pi*(x^2/(4*pi^2))=x^2/(4*pi)}}}


Since the piece used for the circle was x, the perimeter of the square must be 350-x.  Therefore the measure of one side of the square must be {{{(350-x)/4}}}, and then the area of the square must be {{{(350-x)^2/16}}} or {{{(x^2-700x+122500)/16}}}.


We are given that the two areas are equal, so:


{{{x^2/(4*pi)=(x^2-700x+122500)/16}}}


A little manipulation gets us to:


{{{(1-4/pi)x^2-700x+122500=0}}}


which is a quadratic in x with {{{a=(1-4/pi)}}}, {{{b=-700}}}, and {{{c=122500}}}


So, {{{x = (700+-sqrt(490000-4*(1-4/pi)*122500))/2(1-4/pi)}}}


I'll let you deal with this arithmetic horror because I'm WAY too lazy.  But one of the roots will be negative and you can exclude it as extraneous.  Remember the root of this equation is the length of the piece of wire used to make the circle, you have to subtract from 350 to get the length of the wire used for the square.  Good luck.  Unfortunately, the Eighth Amendment to the Constitution of the United States only prevents the Federal Government, not Math Teachers, from inflicting cruel and unusual punishments.