4.5.8-eg-1 - f ( x ) = 2 x + 3000 x , x (0 , ) , where f (...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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 , undened , 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 conrms 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...
View Full Document

This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

Page1 / 2

4.5.8-eg-1 - f ( x ) = 2 x + 3000 x , x (0 , ) , where f (...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online