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a farmer plans to enclose a rectangular pasture adjacent to a river. (see figure). the pasture must contain 80,000 square meters in order to provide enough grass for the herd. what dimensions will require the least amount of fencing if no fencing is needed along the river?

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Therefore, L has a relative and absolute minimum when x = 400 m

What is Perimeter and Area ?

The area is as defined as the amount of space occupied by any shape in two dimensions. Whereas, Perimeter is the boundary or the outline of a flat shape.

Let x = length of the side parallel to the river

     y = length of each of the other two sides

Then xy = 80,000, so y = 80,000/x

Minimize:  L = length of the fence

L = x + 2y

  = x + 160,000/x, where x > 0

L' = 1 - 160,000/x2 = (x2 - 160,000)/x2

L' = 0 when x = 400 or -400

Since x can't be negative, x must be 400.

When 0 < x < 400, L' < 0.  So, L is decreasing.

When x > 400, L' > 0.  So L is increasing.

Therefore, L has a relative and absolute minimum when x = 400 m

                                                                                    y = 80,000/x = 200 m

The fence is shortest if the side parallel to the river has length 400 m and the other 2 sides each have length 200 m.  

To know more about Area, visit:

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