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. How can he do this so as to minimize the cost of the fence?

Answer 1
If the rectangular field has notional sides #x# and #y#, then it has area:
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:

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:

To optimise cost, using the Lagrange Multiplier #lambda#, with the area constraint :
#bbnabla C(bbx) = lambda bbnabla A#
#bbnabla L(bbx) = mu bbnabla A #
#implies mu = 3/y = 2/x implies x = 2/3 y #
#:. qquad qquad {(y = 3*10^3 " ft"),(x = 2* 10^3 " ft"):}#

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

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Answer 2

To minimize the cost of the fence while fencing an area of 6 million square feet in a rectangular field and then dividing it in half with a fence parallel to one of the sides, the farmer should aim to minimize the perimeter of the initial rectangle plus the additional dividing fence.

Let the length of the rectangle be (L) feet and the width be (W) feet, with the area (A = L \times W = 6,000,000) square feet. The perimeter of the rectangle is (P = 2L + 2W), and the total length of fencing needed, including the divider, is (F = P + L = 2L + 2W + L = 3L + 2W).

To minimize fencing, we should find a relationship between (L) and (W) from the area constraint, and then minimize (F).

From (A = L \times W), we get (W = \frac{A}{L} = \frac{6,000,000}{L}).

Substituting (W) in (F = 3L + 2W), we get: [F = 3L + 2\left(\frac{6,000,000}{L}\right) = 3L + \frac{12,000,000}{L}]

To find the minimum fencing needed, we differentiate (F) with respect to (L) and set the derivative equal to zero.

(\frac{dF}{dL} = 3 - \frac{12,000,000}{L^2} = 0)

Solving for (L), we find: [3L^2 = 12,000,000] [L^2 = 4,000,000] [L = 2000 , \text{feet}]

Using (L) to find (W): [W = \frac{6,000,000}{L} = \frac{6,000,000}{2000} = 3000 , \text{feet}]

Thus, to minimize the cost of the fence, the farmer should create a rectangle that is 2000 feet by 3000 feet and then divide it with a fence parallel to the 2000-foot sides. This way, the amount of fencing used is minimized, because this configuration offers the smallest perimeter for the given area, hence the least total length of fencing required when including the divider.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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