Minimizing the Cost of Phone Line Construction
Date: 12/3/95 at 20:58:13 From: Anonymous Subject: optimization-calculus A telephone company has to run a line from point A on one side of a river to another point B that is on the other side, 5km down from the point opposite A. The river is uniformly 12km wide. The company can run the line along the shoreline to a point C and then under the river to B. The cost of the line along the shore is $1000 per km and the cost under the river is twice as much. Where should point C be to minimize the cost? (ANSWER: x=4*the square root of 3) B ---------------! / ! / ! / ! / ! A--------------! C Thank you in advance for your help.
Date: 1/15/96 at 10:7:49 From: Doctor Ethan Subject: Re: optimization-calculus This problem is much like the last one that you sent in. Here is how I would approach it. For the sake of the problem I am considering A to be the point 0,0 B to be 5,12 and C to be x,0 The Total cost TC = x * $1000 + Sqrt[(5-x)(12)] * $2000 Do you see where I got this? The first part x * $1000 is just the cable along the side from A to C The second half is the cable from C to B The Sqrt[(5-x)(12)] is just the Pythagorean theorem to find the length across the river. Now see if you can find the right spot for C. Good luck! -Doctor Ethan, The Geometry Forum
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