Question

You have a wire that is 44 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 1

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

$x+y=44$

$y=44-x$ -----> equation A

step 1

Find out the area of square

The perimeter of square is equal to the length of the first piece

$P=4b$

where

b is the length side of square

$P=x\ cm$

$x=4b$

$b=x/4$

Find the total area A

The area of square is

$A_1=b^2$

$A_1=\frac{x^{2}}{16}$

step 2

Find out the area of the circle

The circumference of the circle is equal to the length of the other piece

$C=(44-x)\ cm$

The circumference is equal to

$C=2\pi r$

so

$2\pi r=(44-x)\ cm$

Find the radius of the circle

$r=\frac{(44-x)}{2\pi}\ cm$

Find the area of the circle

$A_2=\pi r^{2}$

substitute the value of r

$A_2=\pi (\frac{(44-x)}{2\pi})^{2}$

$A_2= \frac{(44-x)^2}{4\pi}$

step 3

Find out the total area

$A=A_1+A_2$

substitute

$A=\frac{x^{2}}{16}+\frac{(44-x)^2}{4\pi}$

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

$x=14.64\ cm\\y=44-14.64=29.36\ cm$

therefore

The circumference of the circle is 29.36 cm