----------------------------------

  0         c                      L





Given L=10m, find c such that  a circle made from the material 0..c and a 

square made from the material c..L  have minimum area.





 A_circle = 



 A_square = 





A_total (= A) = A_circle + A_square = 



Take derivative of A wrt c:

dA/dc =  = 



[ Note that  so the curve is concave up, thus the critical point we are about to find is a local minimum]



Set first deriv. to zero & solve for c:  c = 4.39901m  (or  to one decimal place) 

 

I have separately verified that this value of c minimizes the total area.  You should also verify it to convince yourself.   Also a good idea is to plug in the extreme values of c (0, 10) to gain an understanding of the possible areas.  Here is a graph of A(c) to help you visualize:





  





 

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