Answer:
The circumference of the circle is 29.36 cm
Step-by-step explanation:
Let
x -----> the length of first piece (shape of square)
y ------> the length of the other piece (shape of a circle)
we know that
[tex]x+y=44[/tex]
[tex]y=44-x[/tex] -----> equation A
step 1
Find out the area of square
The perimeter of square is equal to the length of the first piece
[tex]P=4b[/tex]
where
b is the length side of square
[tex]P=x\ cm[/tex]
[tex]x=4b[/tex]
[tex]b=x/4[/tex]
Find the total area A
The area of square is
[tex]A_1=b^2[/tex]
[tex]A_1=\frac{x^{2}}{16}[/tex]
step 2
Find out the area of the circle
The circumference of the circle is equal to the length of the other piece
[tex]C=(44-x)\ cm[/tex]
The circumference is equal to
[tex]C=2\pi r[/tex]
so
[tex]2\pi r=(44-x)\ cm[/tex]
Find the radius of the circle
[tex]r=\frac{(44-x)}{2\pi}\ cm[/tex]
Find the area of the circle
[tex]A_2=\pi r^{2}[/tex]
substitute the value of r
[tex]A_2=\pi (\frac{(44-x)}{2\pi})^{2}[/tex]
[tex]A_2= \frac{(44-x)^2}{4\pi}[/tex]
step 3
Find out the total area
[tex]A=A_1+A_2[/tex]
substitute
[tex]A=\frac{x^{2}}{16}+\frac{(44-x)^2}{4\pi}[/tex]
This is a vertical parabola open upward
The vertex is a minimum
using a graphing tool
The vertex is the point (14.64, 67.77)
see the attached figure
For x=14.64 cm -----> the area is a minimum
The lengths of the wire are
[tex]x=14.64\ cm\\y=44-14.64=29.36\ cm[/tex]
therefore
The circumference of the circle is 29.36 cm
