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February 5th, 2009, 06:39 AM   #1
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Problem involving Chain Rule, Finding Maximum/Minimum point

[color=#400080]here's the problem: Two points, A and B are located 2km and 3km respectively, opposite from a shoreline. What point C on the shoreline could be chosen for Spot X to minimize the sum of the distances from A and B to the Spot X, if the distance between D and E is 4km?

the book says the answer is 8/5. Please show the solution.[/color]


Illus. in this link: http://i172.photobucket.com/albums/w33/ ... _14/ll.jpg
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February 5th, 2009, 04:54 PM   #2
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Re: Problem involving Chain Rule, Finding Maximum/Minimum point

The answer can be found by using the fact that the minimum of a differentiable function, if it is not located at a boundary point, is a point at which the tangent line is horizontal, i.e., .

Using the Pythagorean theorem, we can see that the distances from and to are



The minimum value of the sum of the two distances, if it is not located at a boundary point, will be at a point where

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February 6th, 2009, 09:34 PM   #3
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Let FADG be a straight line such that FD = 3km and DG = 2km,
then AB + BC = GB + BC, which is minimized when CBG is a straight line,
in which case BD/DG = CF/FG, so x = BD = DG(CF/FG) = 2km(4km/5km) = 1.6km.
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