You have a wire that is 38 cm long. you wish to cut it into two pieces. one piece will be bent into the shape of a square. the other piece will be bent into the shape of a circle. let a represent the total area of the square and the circle. what is the circumference of the circle when a is a minimum?

Answer:16.71 cmStep-by-step explanation:GivenLength of wire L=38 cmOne piece is bent in the form of square and another in the form of circlelet x be the length of circle therefore length of square side [tex]\frac{38-x}{4}[/tex]A=total area of square and circleradius of circle [tex]r=\frac{x}{2\pi }[/tex]area of circle [tex]A_c=\pi r^2=\pi \times (\frac{x}{2\pi })^2[/tex]Area of square [tex]A_s=(\frac{38-x}{4})^2[/tex][tex]A=\pi \times (\frac{x}{2\pi })^2+(\frac{38-x}{4})^2[/tex]To get the minimum value of A we get[tex]\frac{\mathrm{d} A}{\mathrm{d} x}=\frac{2x}{4\pi }-\frac{2(38-x)}{16}[/tex][tex]\frac{\mathrm{d} A}{\mathrm{d} x}=0[/tex][tex]\frac{x}{4\pi }=\frac{38-x}{16}[/tex][tex]x=\frac{38\pi }{4+\pi }[/tex]Therefore circumference of circle[tex]x=\frac{38\pi }{4+\pi }=\frac{119.396}{7.142}=16.717 cm[/tex]