length and width of area?

Question

A rancher wants to fence in an area of 580 square feet in a rectangular field using fencing material costing 1.4 dollars per foot, and then divide it in half down the middle with a partition, parallel to one side, constructed from material costing 0.6 dollars per foot.
Assuming that the partition is parallel to the side which gives the width of the field, find the dimensions of the field of the cheapest design.

Answer 1

Let x = width
Then length = 580/x
Cost = C = 2(x +580/x)(1.4) +.6x
C = 3.4x + 1624/x
dC/dx = 3.4 -1624/x^2 = 0
x = sqrt (1624/3.4) = 21.856 feet = width
580/x = 26.538 feet = length

Answer 2

wl = 580 => l = 580/w
C(w) = 1.4(2w+2l) + 0.6w = 2.8(w+580/w) + 0.6w
C'(w) = 2.8(1-580/w^2) + 0.6 = 0
w = 21.86 ft
l = 26.54 ft