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4.5.8-eg-1

# 4.5.8-eg-1 - f x = 2 x 3000 x x ∈(0 ∞ where f x =...

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Example A farmer wants to build a rectangular pen which will be bounded on one side by a river and on the other three sides by a fence. If the pen is to enclose 3000 square feet of area, find the dimensions of the pen which use the least amount of fence. Solution: Let y denote the length of the fence (in ft) on the side of the pen parallel to the river, and let x be the length (in ft) of the fence on each side of the pen perpendicular to the river. The goal is to minimize the amount of fencing required, which in this case is a total of 2 x + y (1) feet. To rewrite this expression as a function of one variable, we can use the fact that the area inside the fence must be 3000 square feet, or that xy = 3000 = y = 3000 x . (2) So from (1), the total amount of fencing required is f ( x ) = 2 x + 3000 x , x > 0 , written as a function of x alone. Note that any x (0 , ) is a valid length for the designated side of the pen and that the value of the corresponding side length y is computed from x using (2). The goal then is to minimize the function

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Unformatted text preview: f ( x ) = 2 x + 3000 x , x ∈ (0 , ∞ ) , where f ( x ) = 2-3000 x 2 = 2 x 2-3000 x 2 = ( , at x = ± √ 1500 = ± 10 √ 15 , undeﬁned , at x = 0 . But only the point x = 10 √ 15 is in the given domain of f , so this is the only critical point of f . Using f 00 ( x ) = 6000 x 3 , it follows that f 00 (10 √ 15) = 6000 10 3 15 3 / 2 > , which conﬁrms that the function f has a local minimum at x = 10 √ 15 from the Second Derivative Test. Note also that f 00 ( x ) > , for all x ∈ (0 , ∞ ) , so that the graph of f ( x ) is concave up over its entire domain. From this it follows that the local minimum is an absolute minimum over this domain. Thus the optimal value of x is x = 10 √ 15 and the corresponding value of y is y = 3000 10 √ 15 = 300 · √ 15 15 = 20 √ 15 . The pen using the least amount of fencing has dimensions 10 √ 15 ft × 20 √ 15 ft . 2...
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4.5.8-eg-1 - f x = 2 x 3000 x x ∈(0 ∞ where f x =...

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