A 5,000 m² rectangular area of a field is to be enclosed by a fence, with a moveable inner fence built across the narrow part of the field, as shown.The perimeter fence costs $10/m and the inner fence costs $4/m. Determine the dimensions of the field to minimize the cost to construct the fencing.

Answer 1

let the width of the field be x m
and the length of the field be y m

so we need 2x + 2y for the perimeter + x for the inner

we know xy = 5000
y = 5000/x

Cost = 10(2x+2y) + 4x = 24x + 20y
= 24x + 20(5000/x)

dCost/dx = 24 - 100000/x^2 = 0 for a minimum cost
24x^2 = 100000
x^2 = 12500/3
x = appr 64.55 m
y = appr 77.46 m