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04132010  #1 
Join Date: Apr 2010
Posts: 1

Optimization Problem
What would be the approach for this problem:
A power house, P, is on one bank of a straight river 200m wide, and a factory, F, is on the other bank 400m downstream from P. The cable has to be taken across the river under water at a cost of $12.00/m. On land the cost is $6.00/m. What path should be chosen so the cost is minimized? Any help would be much appreciated. 
04152010  #2  
Join Date: Dec 2008
Posts: 249

Quote:
L = distance on land Cost = 12*W + 6*L If the cable is run underwater in a straight line to the opposite bank to some point between the factory and a point straight across from the power house, then the distance underwater is the hypotenuse of the right triangle with legs 200m and 400L meters. W^2 = 200^2 + (400L)^2 W = sqrt(40000 + 160000  800*L + L^2) W = sqrt(200000  800*L + L^2) Now plug W back into the cost equation. Cost = 12*sqrt(200000  800*L + L^2) + 6*L One way to find the minimum cost is to take the derivative of cost set it equal to 0, and solve for L. d/dL(cost) = (6*sqrt(200000  800*L + L^2) + 12*L  4800)/sqrt(200000  800*L + L^2) Setting the der of cost to 0 gives: 0 = 6*sqrt(200000  800*L + L^2) + 12*L  4800 (4800  12*L)/6 = sqrt(200000  800*L + L^2) Squaring both sides gives: (144*L^2  115200*L + 23040000)/36 = L^2  800*L + 200000 36*L^2  28800*L + 7200000 = 144*L^2  115200*L + 23040000 0 = 108*L^2  86400*L + 15840000 Using quadratic formula gives L = 284.53 or L = 515.47 Throw away the value greater than 400 and plug the value for L back into the W question to solve for W. W = sqrt(40000 + (400  284.53)^2) = 230.94 Min cost = $4478.46. The cable should be run to a point 284.53m from the factory on the opposite bank of the river. 

04192010  #3 
Join Date: Mar 2005
Posts: 10,647

Thank you for that detailed explanation.
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04202010  #4 
Join Date: Feb 2010
Posts: 117

yes
MAS1 that was a rather thorough explanation I'm getting to understand these kind of problems better now

04232010  #5 
Join Date: Feb 2010
Posts: 388

Same, Gotten alot better since i've looked over your solutions to the problems.
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09222010  #6 
Join Date: Mar 2010
Posts: 46

Same here

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