A farmer plans to fence a rectangular pasture as two adjacent pens of the same s

Question

A farmer plans to fence a rectangular pasture as two adjacent pens of the same size next to a river. The pasture must contain 270,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river? Use the methods of Calculus 1.

Explanation / Answer

So you have two smaller adjacent rectangles. Let x be the length of the horizontal side of a smaller rectangle and y be the vertical side length. Since the area of the two rectangles is 270,000, then we have 2xy = 270,000 => xy = 135,000 => y = 135,000/x Now we must minimize the amount of fencing. Thus Amount of fencing = 2x + 3y = 2x + 3(135,000/x) = 2x + 405,000/x Taking the derivative and setting it equal to 0: 0 = 2 - 405,000/x^2 2 = 405,000/x^2 2x^2 = 405,000 x^2 = 202,500 x = 450 So if 2xy = 270,000 , then 2(450)y = 270,000 900y = 270,000 y = 300 Therefore the least amount of fencing required if no fencing is needed along the river is: 2(450) + 3(300) = 900 + 900 = 1800m

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