Optimization question

Question

Dazzline and Nihad wants to raise chickens in a rectangular field with three internal partitions as shown. If the area of the field must be 19200 square feet, what dimensions will require the least fencing?

(Note that the internal partitions also require fencing.)

Answer 1

area = 19200= Lw

L =19200/w

2 Lengths

3 internal + 2 outside widths

5w + 2L = perimeter = total fencing required

P = 5w + 2(19200/w)

P'(w)= 5 - 38400/w^2 =0

5w^2 -38400 = 0

w^2 = 38400/5

w^2 = 7680

w = sqr7680= 16sqr30 =87.64 feet wide

L = 19200/87.64 = 219.09 feet Long

219.09(87.64) = 19200.96

5(87.64) +2(219.09) =438.62 + 438.18= 876.8 feet minimum fencing required

minor rounding errors/estimates