Question

Question 2 options: A fence must be built to enclose a rectangular area of 45,000 ft2. Fencing material costs $4 per foot for the two sides facing north and south and $8 per foot for the other two sides. Find the length and width that will produce the least expensive fence.

Answer 1

Answer:

The length that would produce the least expenses is  $A= 300 ft$

The width  that would produce the least expenses is  $B= 150 ft$

Step-by-step explanation:

From the question we are told that the

           Required area to enclose is $A = 45,000 ft^2$

            Fencing material cost is $C =$$4 per foot for north and south

            Fencing material cost for east and west $C_{E/W} =$ $8

The diagram for this question is shown on the first uploaded image

      From the diagram is mathematically evaluated as

                             AB = 45000

                    =>        $B = \frac{45000}{A}$

The overall cost of building this fence is

                $T = 2 A (4) + 2(B)(8)$

Substituting for B in the equation above

                $T = 8A +16 (\frac{45000}{A} )$

differentiating both sides with respect to x

              $T' (A) = 8 - \frac{16 *45000}{A^2}$

At minimum possible cost  $T'(A) = 0$

              => $8 - \frac{16 *45000}{A^2} = 0$

                  $8A^2 = 16*45000$

                   $A^2 = \frac{16*45000}{8}$

                    $A = \sqrt{\frac{ 16*45000}{8} }$

                      $= 300 ft$

Then B is mathematically evaluate as

                 $B = \frac{45000}{300}$

                     $= 150 ft$

Then the maximum is mathematically evaluated as

                $T = 8 (300) + 16(150)$

                     =$4800