A 3m length of wire has to be bent into the form of a rectangle with an external circular loop at one corner such that the rectangle?

Let the Length and breadth  of the rectangle be ----  2L m and L m respectively.

Also Let Radius of the circular loop = R m

Both of these shapes have been made out of 3 m of a wire.

Hence  3  =  6 L + 2 π R

=>  6 L +  2 π R  =  3  ................... (1)

=>  R  =  ( 3 - 6 L )/ 2 π     ....................... (2)

Total area ( A ) of the two shapes = 2 L² + π R²   ................... (3)

A  =  2 L² + π R²  

 A  = 2 L² + π [ ( 3 - 6 L )/ 2 π ]² =    2 L² +  (9/4π²) ( 4 L² - 4 L + 1 )

=> dA/dL  =  4 L +  (9/4π²) [ 8 L - 4 ]  =  4 L + (9/π²) ( 2 L - 1 )

=> d²L/dL²  =   4 + (18/π²)  =  a +ve number

For the total area being minimum, dA/dL  =  0

=>  4 L + (9/π²) ( 2 L - 1 )  =   0

Hence  L  =  (9/2) * 1/(2π²-9 ) =  0.419 m

from (2) we have R = 0.411 m

Hence Dimension of -

Rectangle --  0.419 m x 0.838 m........... nswer

Circle Radius ( R )  = 0.411 m ............ Answer