A piece of wire 40 cm long is to be cut into two pieces. One piece will be bent to form a circle and; the other will be bent to form a square. Find the lengths of the two pieces that cause the sum of the area of the circle and the area of the square to be a minimum. The perimeter of the square is cm The circumference of the circle is cm

Step-by-step explanation:

The total length of the wire  = 40 cm

Let us assume the circumference of the circle  = k cm

So, the circumference of the square  = ( 40 - k) cm

Now,as circumference of circle = 2πR

⇒2πR = k cm    or, R  =

Area  of the  circle = πR²  

or, A  =  

    ....... (1)

Similarly , perimeter of square  = 4 x Side

⇒ 4 x Side  =  ( 40 - k) cm    ⇒ S =

Area  of the  square = (Side)²    =

 Solving, we get:     ....... (1)

So, combining (1) and (2), total area A is

or, we get: A  = 0.07962 k²  + 0.0625 k² + 100 - 5 k

A  =  0.1422 k² - 5 k + 100

For axis of symmetry :  

⇒ k = 17.6 inches

So, the perimeter of circle  = 17.6 cm

Perimeter of square  = (40 - 17.6 )  = 22.4 cm

in mathematics, a function is a relationship between two variables such that the value of one of the variables, called the dependent variable, is determined by the value of the other variable, called the independent variable. there is a specific term we use to describe the set of independent variables of a function.