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My home is $2m$ South from the river.If I go $12m$ straight to the West from my home and $1m$ straight to the South then I find my grandmother's house.What is the minimum length to go to grandmother's house taking water from river.
I can not understand what can I do with this problem.This is my homework.Please help me anyone

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    $\begingroup$ If the river bank is straight, then think about relflecting your house in it. $\endgroup$
    – Henry
    Commented Jan 9, 2014 at 18:05

2 Answers 2

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Mirror the house of your grandma at the river, connect your home to the mirrored house, the intersection of that line with the river is the point to get water from. Move only in straight lines.

By the way: This is the way that light travels (assuming that the river is a real mirror). Meaning, light takes the shortest way

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Assuming you don't know the "mirror trick", denote the point at which you take water from the river by $x$. Then the total distance is: $$y = \sqrt{2^2+x^2} + \sqrt{3^2+(12-x)^2}$$ Now, to find the minimum, find out when $dy/dx = 0$...

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