3. From a tract of land a farmer plans to fence a rectangular region and then di

Question

3. From a tract of land a farmer plans to fence a rectangular region and then divide it into two identical sized rectangular lots by putting a fence down the middle. The tract of land is adjacent to a road, but only one of the subdivisions will use the road as a boundary. The farmer wants to use sturdy fencing adjacent to the road, and this costs $35 per metre. For each of the other exterior fences the farmer will use standard fencing which costs $25 er metre. For the interior fence between the two subdivisions the farmer will use light fencing which only costs $20 per metre (the interior fence is parallel to the road). Find the dimensions of each lot to minimise the total cost for fencing if the total area to be enclosed is 400,000 square metres (Assume that the road runs in a straight line.)

Explanation / Answer

Let the northern part of the fence i.e, breadth = X

and, length of the western and eastern pieces of the fence = Y

Therefore, 35X + 25Y + 20Y = 400000 ................equation ''1''

It implies, Y = (400000 -35X) / 45

Then, Area of fence = length * breadth

= [ (400000 - 35X) / 45 ] * [ X ]

= ( 400000X - 35X^2 ) / 45

It implies, = ( 400000 / 45) - (70X / 45) = 0 [It is so because a derivative is taken]

= ( 400000 - 70X ) / 45 = 0

= 400000 - 70X = 0

Therefore, X = 5714.29 metres

By putting value of 'X' in equation ''1''................

35(5714.29) + 45Y = 400000

It implies, 45Y = 400000 - 200000

Therefore, Y = 4444 metres (approx.)

Hence, the dimensions of each lot to minimize the total cost for fencing if the total area to be enclosed is 400000 square metres are:-

Lenghth = 4444 m (approx)

Breadth = 5714.29 m (approx)

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