Question 1114168
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The problem asks to minimize the sum  2*(3x + 6y)  under the condition  xy = 45000.


It is the same as to minimize  the form  x+2y  under the condition  xy = 45000.


Then  x + 2y = {{{x + 2*(45000/x)}}} = {{{x + 90000/x}}}.


To find the minimum of this function of x, take the derivative and equate it to zero:


{{{1}}} - {{{90000/x^2}}} = {{{(x^2-90000)/x^2}}} = 0,


and the root (the solution) is   x = 300.


<U>Answer</U>.  The dimensions  x= 300 ft (north and south sides)  and  {{{45000/300}}} = 150 ft  (other two sides)  give the required minimum.


         The minimum cost of the fence is  2*(3*300 + 6*150) = 3600 dollars.
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