Answer:16.71 cm
Step-by-step explanation:
Given
Length of wire L=38 cm
One piece is bent in the form of square and another in the form of circle
let x be the length of circle
therefore length of square side [tex]\frac{38-x}{4}[/tex]
A=total area of square and circle
radius 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]
