Rancher Optimization

Question

A rancher intends to mark off a rectangular plot of land that will have an area of 1500 m^2. The plot will be fenced and divided into two equal proportions by additional fence parallel by two sides. Find the dimensions of the land that require the least amount of fencing

Answer 1

Area of Rectangular Plot = L*W = 1500

Total Fencing = Perimeter + width = 2*L + 3*W

Rewrite first equation as L = 1500/W and put into second equation

Total Fencing T = 2*1500/W + 3*W

T = 3000/W + 3*W

T' = -3000/W^2 + 3

0 = 3 - 3000/w^2

3000/W^2 = 3

3*W^2 = 3000

W^2 = 1000

W = √1000 m = 10√10 m

L = 1500/10√10 m = √15√10 m