Question

A rancher plans to set aside a rectangular region of one square kilometer for cattle and wishes to build a wooden fence to enclose the region. Since one side of the region will run along the road, the rancher decides to use a better quality wood for that side which costs three times as much as the wood for the other sides. What dimensions will minimize the cost of the fence?

Answer 1

x y = 1

c = 3 x + x + 2 y = 4 x + 2 y

c = 4 x + 2/x

dc/dx = 4 - 2/x^2
= 0 for minimum

4 = 2/x^2
x^2 = 1/2
x = 1/sqrt 2 = sqrt 2/2 along road
y = sqrt 2