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Thread: A road between two cities

  1. #1

    A road between two cities

    I appreciate any help that I get, I just can't figure out how to do this problem. If anyone else knows, I would love some input!

    Two cities lie south of a straight road but they are not connected to the road or to each other. The people of the city decided to build two roads, one from each city to the existing road. We do not have exact values to work with, so you need to assign letters to the distances you need for the problems. Distance from one city to the road; distance from other city to the road; Distance on the road they are from each other. The answer will be in the terms of these three unknown constants.

    Use Calculus - Optimization - to find the shortest route connecting these two cities via the existing road.

    Here is a picture of the problem >>
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  2. #2
    Eater of Worlds
    galactus's Avatar
    Jul 2006
    Chaneysville, PA
    I f I am interpreting correctly, you must find a point P which minimizes the distances from the cities to the main road.

    You can do this with similar triangles.

    $\displaystyle \frac{x}{a}=\frac{c-x}{b}$

    Solve for x and we get $\displaystyle x=\frac{ac}{a+b}$

    You could derive a formula using Pythagoras and then differentiate, set to 0 and solve for x, but you'll get the same solution only with a lot more work.

    $\displaystyle D=\sqrt{a^{2}+x^{2}}+\sqrt{b^{2}+(c-x)^{2}}$

    $\displaystyle D'=\frac{x\sqrt{x^{2}-2cx+b^{2}+c^{2}}+\sqrt{x^{2}+a^{2}}(x-c)}{\sqrt{(x^{2}+a^{2})(x^{2}-2cx+b^{2}+c^{2})}}$

    $\displaystyle x\sqrt{x^{2}-2cx+b^{2}+c^{2}}+\sqrt{x^{2}+a^{2}}(x-c)=0$

    $\displaystyle x^{2}(x^{2}-2cx+b^{2}+c^{2})=(x^{2}+a^{2})(x-c)^{2}$

    $\displaystyle (b^{2}-a^{2})x^{2}+2a^{2}cx-a^{2}c^{2}=0$


    $\displaystyle ((a-b)x-ac)(ac-(a+b)x)$

    $\displaystyle (a-b)x-ac=0; \;\ x=\frac{ac}{a-b}$

    $\displaystyle ac-(a+b)x=0; \;\ \boxed{x=\frac{ac}{a+b}}$
    Last edited by galactus; Nov 24th 2008 at 05:38 AM.
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