Question 1152181
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  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 = {{{ pi(r^2) = pi*(c/(2pi))^2 =  c^2/(4pi) }}}

 A_square = {{{ ((L - c)/4)^2 =  (10-c)^2/16 }}}


A_total (= A) = A_circle + A_square = {{{ c^2/(4pi) + (10-c)^2/16 }}}

Take derivative of A wrt c:
dA/dc = {{{ 2c/(4pi) + 2(10-c)(-1)/16 }}} = {{{ c/(2pi) + (c-10)/8 }}}

[ Note that {{{d^2A/dc^2 = (1/(2pi)) + (1/8) > 0 }}} 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 {{{highlight(4.4m)}}} 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:


{{{ graph(400,400,-1,10,-1,10, x^2/(4*pi)+(10-x)^2/16) }}}