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?

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

-----> equation A

step 1

Find out the area of square

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

where

b is the length side of square

Find the total area A

The area of square is

step 2

Find out the area of the circle

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

The circumference is equal to

so

Find the radius of the circle

Find the area of the circle

substitute the value of r

step 3

Find out the total area

substitute

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

therefore

The circumference of the circle is 29.36 cm

hello,

as we don't know if g(x) is continue,

answer b

f(x)=2x²-8x+1=2(x²-4x+4)+1-8=2(x-2)²-7

vertex is (2,-7) ( a minimum since coefficient of x² is > 0)

hello,

as   long as they add up to 180 its correct.