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?

Use half the fencing in each direction: 3 segments in one direction, 2 in the other.

W * L = 24 million
3 W = 2 L

W * 3 W / 2 = 24 million
W^2 = 16 million
W = 4000
L = 6000

This shortcut (3 W = 2 L)
is derived from total fencing = 3W + 2L
and W*L = 24 million
total fencing = 3W + 2 * 24 million / W
and taking the derivative, setting to 0, and solving for W

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Cricetus Cricetus · Answer 2 · 2021-10-26T18:13:37+02:00

The lengths of sides of the rectangular field will be "6 and 4".

Let,

Length = x
Width = y

Area of field,

→ [tex]A = xy[/tex]

[tex]xy = 24[/tex]

[tex]y = \frac{24}{x}[/tex]...(equation 1)

Required fencing will be:

→ [tex]S = 2x+3y[/tex]

[tex]= 2x+\frac{3\times 24}{x}[/tex]

[tex]= 2x+\frac{72}{x}[/tex]...(equation 2)

Differentiate with respect to "x", we get

→ [tex]\frac{dS}{dx} = 2-\frac{72}{x^2}[/tex]

To minimize the cost of fence, we get

→ [tex]\frac{dS}{dx} =0[/tex]

[tex]2-\frac{72}{x^2} =0[/tex]

[tex]x^2=36[/tex]

[tex]x =6 \ million \ feet[/tex]

and,

→ [tex]y=\frac{24}{6}[/tex]

[tex]=4[/tex]

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