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Math Help - Distance

  1. #1
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    Red face Distance

    I dont quite get the picture of this question....

    Aruneem travels from Patna to Calcutta, a distance of 200 km at the speed of 40 km/hr . At the same time, Nilanjan starts from Calcutta at a speed of 20 km/hr along a road, which is perpendicular to the road on which Aruneem is travelling. When will Aruneem and Nilanjan be closest to each other?

    ans: In 4 hr
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  2. #2
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    Hello, MathLearner!

    Aruneem travels from Patna to Calcutta, a distance of 200 km at the speed of 40 km/hr.
    At the same time, Nilanjan starts from Calcutta at a speed of 20 km/hr along a road
    which is perpendicular to the road on which Aruneem is travelling.
    When will Aruneem and Nilanjan be closest to each other?

    Answer: in 4 hours
    Code:
               40t    A     200-40t
        P * - - - - - * - - - - - - - - * C
                         *              |
                            *           |
                               *        | 20t
                                  *     | 
                                     *  |
                                        * B

    Aruneem starts at P and travels toward C at 40 km/hr.
    . . After t hours, he has travelled 40t km to point A.
    Note that: . AC \:=\:200-40t

    At the same time, Nilanjan starts at C and travels south at 20 km/hr.
    . . After t hours, he has travelled 20t km to point B.

    Their distance is AB, the hypotenuse of right triangle ACB.

    Hence: . D \;=\;AB^2 \:=\:(200-40t)^2 + (20t)^2 \:=\:2000t^2 - 16,000t + 40,000

    Then: . D' \;=\;4000t - 16,000 \:=\:0 \quad\Rightarrow\quad t \,=\,4


    Therefore, minimum distance occurs in 4 hours.

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  3. #3
    Member Jonboy's Avatar
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    soroban i understood everything except where you got the D' equation.
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  4. #4
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    Hello, Jonboy!

    We have right triangle ABC with C = 90^o,\;AC \:=\:200-40t,\;BC \:=\:20t
    . . and we want to minimize AB.

    Pythagorus says: . AB^2 \;=\;AC^2 + BC^2 \;=\;(200-40t)^2 + (20t)^2

    See it?

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  5. #5
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    hello again Sorobon !!!

    Brilliant answer.. now i understood what exactly the question meant...

    Thanks
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  6. #6
    Member Jonboy's Avatar
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    Thanks for trying to explain Soroban, I see what the picture means. But as far as getting the D prime equation, that's not something I've learned and I don't see how you got there.

    But if I wanted to find the minimum here, I would've just done:

    \frac{ - b}{2a} = \frac{16000}{4000} = 4
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  7. #7
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    Isomorphism's Avatar
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    Quote Originally Posted by Jonboy View Post
    soroban i understood everything except where you got the D' equation.
    He differentiated.
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  8. #8
    Member Jonboy's Avatar
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    Quote Originally Posted by Isomorphism View Post
    He differentiated.
    So wish I knew how to do that. I will next year when I take Calc. Maybe sooner if I can find a simple explanation of Calc at my library.
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  9. #9
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Jonboy View Post
    So wish I knew how to do that. I will next year when I take Calc. Maybe sooner if I can find a simple explanation of Calc at my library.
    Differentiation is just finding the equation that will give slope at a point. It is defined as

    Let f(x) be a differentiable function.

    Then \frac{dy}{dx}=f'(x)=\lim_{\Delta{x}\to{0}}\frac{f\  left(x+\Delta{x}\right)-f(x)}{\Delta{x}}
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  10. #10
    Member Jonboy's Avatar
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    Cool

    calculus has such a beautiful notation. thanks for explaining Mathstud28 to this lost person.
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