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 July 27th, 2012, 08:39 PM #1 Newbie   Joined: Jul 2012 Posts: 1 Thanks: 0 Cost minimizing function. You wish to make a rectangular pen with fence which faces in the north-south direction and in the east-west direction. The pen is to hold 200 square feet. The north-south fence costs $2 per square foot. The east-west fence costs$1 per square foot. You make a pen that minimizes the cost. The perimeter of the pen in feet is Possible answers: 20 30 50 60 40
 July 27th, 2012, 08:59 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Cost minimizing function. Let x be the length of each of the two sides of the pen that run east-west and y be the length of each of the two sides of the pen that run north-south. Hence, the cost function in dollars is (the cost of the fencing should be given in dollars per linear foot, not square foot): $C(x,y)=1(2x)+2(2y)=2x+4y$ From the statement "The pen is to hold 200 square feet" we have from the formula for the area of a rectangle: $A(x,y)=xy=200$ Hence: $y=\frac{200}{x}$ Now, substituting this into the cost function, we get a function in one variable, namely x: $C(x)=2x+4$$\frac{200}{x}$$=2x+800x^{\small{-1}}$ Now, we may equate the first derivative to zero to find the critical value(s): $C'(x)=2-800x^{\small{-2}}=\frac{2x^2-800}{x^2}=\frac{2}{x^2}(x+20)(x-20)=0$ The only positive critical value is $x=20$. We find $C''(20)>0$ thus we have a minimum. Hence: $x=20,\,y=10$ and the perimeter P is: $P=2(x+y)=2(20+10)=60$
 July 28th, 2012, 01:16 AM #3 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Cost minimizing function. Another approach: Lagrange multipliers. $C(x,y)=2x+4y$ subject to the constraint $xy-200=0$. Taking the appropriate partials, this gives rise to the system: $2=\lambda y$ $4=\lambda x$ This implies: $x=2y$ Hence: $2y^2=200\:\therefore\:y=10,\,x=20$
 July 30th, 2012, 04:55 PM #4 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Cost minimizing function. By the way, St. Augustine is the oldest city in the United States continuously inhabited by Europeans and their decendants. Taos, New Mexico, has been continously inhabited for far longer.
 July 30th, 2012, 10:02 PM #5 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Cost minimizing function. Very interesting!

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