Question

A farmer wants to fence an area of 6 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. Let y represent the length (in feet) of a side perpendicular to the dividing fence, and let x represent the length (in feet) of a side parallel to the dividing fence. Let F represent the length of fencing in feet. Write an equation that represents F in terms of the variable x.

Answer 1

If the rectangular field has notional sides X and y then it has area
A(x) =xy { =6•10^6 sq ft }
The length of fencing required, if
x
is the letter that was arbitrarily assigned to the side to which the dividing fence runs parallel, is:l (x) = 3x +2y

It matters not that the farmer wishes to divide the area into 2 exact smaller areas.
Assuming the cost of the fencing is proportional to the length of fencing required, then
C(x)=a L (x)

To optimise cost, using the Lagrange Multiplier
λ
, with the area constraint :

So the farmer minimises the cost by fencing-off in the ratio 2:3, either-way

Answer 2

The relationship between variable F and x is 2x + 12000000/x

Let y represent the length (in feet) of a side perpendicular to the dividing fence, and let x represent the length (in feet) of a side parallel to the dividing fence.

Let F represent the length of fencing in feet.

Area of fencing = 6000000 ft²

Area of fencing = x * (y + y) = x * 2y

Area of fencing = 2xy

6000000=2xy

xy = 3000000

y = 3000000/x

The perimeter is:

F = x + y + y + x + y + y

F = 2x + 4y

F = 2x + 4(3000000/x)

F = 2x + 12000000/x

Therefore the relationship between variable F and x is 2x + 12000000/x

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