a farmer wants to build a fence around a rectangular field of 180 000 m squared and divide it into thirds with fence parallel to one of the sides of the rectangle. what should be the dimensions of each rectangular pen be if the farmer wants to use the least amount of fencing possible?

Answer 1

To minimize the amount of fencing required, the rectangular field should be divided into three equal parts by fences parallel to one side of the rectangle.

Let's assume the length of the rectangular field is L meters and the width is W meters.

If the field is divided into thirds, each rectangular pen would have an area of 180,000 / 3 = 60,000 m².

Let's suppose the length of each pen is 3L meters. Since the pen occupies one-third of the total area, its width would be 60,000 / 3L = 20,000 / L meters.

To minimize the amount of fencing, we need to find the values of L and W that minimize the perimeter of the fence.

The perimeter of the fence would be:
Perimeter = 2L + W + 4(20,000 / L)

We can differentiate the equation with respect to L and set the derivative equal to zero to find the minimum:

d(Perimeter)/dL = 2 - 4(20,000 / L^2) = 0

Simplifying the equation, we get:
2L^2 - 80,000 = 0
L^2 = 40,000
L ≈ 200

Substituting the value of L back into the equation for the width of each pen:

W = 20,000 / L ≈ 20,000 / 200 ≈ 100

Therefore, each rectangular pen should have dimensions approximately 200 meters in length and 100 meters in width in order to use the least amount of fencing possible.