Question 1150956
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From the context, the dimensions of the rectangle are not given for advance - they are unknowns and they should be found

from the minimum cost condition.


Let x be one dimension and y be the other dimension of the rectangle.


Then the cost of the outside perimeter fence is 1.50*(2x+2y) dollars = 3*(x+y) dollars,

while the cost of the fence down middle is 0.50*x dollars.


    Note, that I don't know now, which dimension will be the length and which be the width.

    When the problem will be solved, the solution will tell me it . . . 


So, I need minimize the function  

        f(x,y) = 3*(x+y) + 0.5x     (1)

under the condition

        x*y = 1050.                 (2)



From (2), express  y = {{{1050/x}}}  and substitute it into (1). You will get

        g(x) = {{{3*(x+ 1050/x) + 0.5x}}} = {{{3x}}} + {{{3150/x}}} + {{{0.5x}}}.


Differentiate it over x

        g'(x) = {{{3}}} - {{{3150/x^2}}} + {{{0.5}}}

and equate the derivative to zero.  You will get


        3.5 = {{{3150/x^2}}},    or

        3.5x^2 = 3150

           x^2 = {{{3150/3.5}}} = 900,

           x   = {{{sqrt(900)}}} = 30.


Thus the dimensions of the rectangle are 30 ft  and  {{{1050/30}}} = 35 ft.

The fence down middle has the length of 30 ft;  hence, it is parallel to the shorter side of the rectangle.
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Solved.


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If you want to see many other similar solved problems, look into the lesson 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/Calculus-optimization-problems.lesson>Calculus optimization problems</A>

in this site.