Question

A farmer wants to fence an area of 24 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence? smaller value? and larger value?

Answer 1

area= 24,000,000 = Length*Width
ergo, w=24,000,000/L
Perimeter= 3L+2w Because there is another fence in the middle of the field.
P= 3(24,000,000/w)+2w
\[P \prime =3(24,000,000)(-1)(w ^{-2})+2)\]
\[P \prime =0=\frac{-72,000,000} {w ^{2 }}+2\]
\[-2=\frac{-72,000,000} {w ^{2 }}\]
\[\frac{-72,000,000} {-2} =w ^{2 }\]
\[\sqrt{36,000,000} =\sqrt{w ^{2 }}\]
\[6,000=w\]
therefore....
A=24,000,000=(6,000)(L)
L=(4,000)

Min = 4,000
Max= 6,000