Question

A rectangular field is to be enclosed on four sides with a fence. Fencing costs $2 per foot for two opposite sides, and $3 per foot for the other two sides. Find the dimensions of the field of area 830 ft2 that would be the cheapest to enclose.

Answer 1

Dimension for cheap enclose = 32.45 ft x 23.52 ft

Explanation:

Area of rectangular field, A = 830 ft²

Length = l

Width = w

So we have

                 l x w = 830

                    $l=\frac{830}{w}$

Fencing costs $2 per foot for two opposite sides, and $3 per foot for the other two sides.

            Cost for fencing, C = 2 x 2 x w + 3 x 2 x l = 4 w + 6 l

             $C=4w+6\times \frac{830}{w}$

For minimum cost we have derivative is zero

           $dC=4\times dw-6\times \frac{830}{w^2}\times dw\\\\0=4\times dw-6\times \frac{830}{w^2}\times dw\\\\w^2=1245\\\\w=32.45ft\\\\lw=830\\\\l\times 32.45=830\\\\l=23.52ft$

Dimension for cheap enclose = 32.45 ft x 23.52 ft