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You have a wire that is 77 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?

You have a wire that is 77 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 sh class=

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Answer:

23.62 cm

Step-by-step explanation:

The question asking for a condition when the total area of the square and the circle(A) is minimum/lowest. To find the lowest area you can get, you have to differentiate the formula for the total area(A).

The wire is 77cm long, let's say x is the square side and r is the circle radius. Then it will be

77cm= 4x +    2 *  π  * r  

4x= 77cm -   2 *  π  * r  

x= (77cm -  (π  * r ))/4  

Square area is x^2 while circle area is  π *r^2. Total area will be:

A= square area + circle area

A==  x^2 + π *r^2  

A=  ( (77 -  (π  r ))/4   ) ^2                       +  π *r^2    

A=(5929- (144 *  π  r )  + π^2  r^2)/ 16  + π *r^2    

A= 370.56 - 9  π  r   +   π^2  r^2/16  +  π *r^2      

A= 370.56 - 28.26  r   +  0.616r^2  +  3.14*r^2  =

A= 370.56 -  28.26   r   +  3.756  r^2        

Differentiate the equation to find the lowest point

370.56 -  28.26   r   +  3.756  r^2

- 28.26   +  7.512 r = 0

r=  28.26  / 7.512

r = 3.76 cm

Radius of circle when A minimum is 3.76cm, then the perimeter will be: 2 *  π  *3.76=  23.62 cm

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