Question

A rancher wishes to build a fence to enclose a rectangular pen having area 24 square yards. Along one side the fence is to be made of heavy duty material costing $6 per yard, while the remaining three sides are to be made of cheaper material costing $3 per yard. Determine the least cost of fencing for the pen.

Answer 1

Answer:

The  value  is  $C \approx \ 76$

Step-by-step explanation:

From the question we are told that

   The  area of the rectangular pen is  $A = 24 \ yard^2$

     The  cost of material used to make one side is  $z = \ 6$

    The  cost of material used to make the other sides is  $r = \ 3$

Now  , the fence to be build around the rectangular pen  has four sides, the first opposite sides are equal, let assume each of the to be x yard   and the other opposite sides are also equal as well let assume of the to be y yard

So the cost is mathematically represented as

       $C = zx + r (x + 2y )$

=>   $C = 6x + 3(x + 2y)$

=>   $C = 9x + 6y$

Now the area of the fence is mathematically represented as

      $A = x* y = 24$

=>    $y = \frac{24}{x}$

=>  $C = 9x + 6[\frac{24}{x} ]$

=>   $C = 9x + [\frac{144}{x} ]$

Now differentiating

     $C' = 9 + 144* (-2) x^{-2}$

     $C' = 9 - 288x^{-2}$

At minimum $C' = 0$

So  

     $9 - 288x^{-2} = 0$

     $x^{-2} = 0.03125$

    $x = \sqrt{\frac{1}{0.03125} }$

    $x = 5.66$

Now substituting for x in the equation above to obtain minimum cost

       $C = 9(5.66) + [\frac{144}{5.66} ]$

       $C \approx \ 76$