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Thread: Finding the distance between two points

  1. #1
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    Finding the distance between two points

    Let Q = (0,5) and R = (10,6) be given points in the plane. We want to find the point P = (x,0) on the x axis such that the sum of distances PQ+PR is as small as possible.
    To solve this problem, we need to minimize the following function of x: f(x) = ?
    over the closed interval [A,B] where A = ? and B = ?
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  2. #2
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    Quote Originally Posted by derekjonathon View Post
    Let Q = (0,5) and R = (10,6) be given points in the plane. We want to find the point P = (x,0) on the x axis such that the sum of distances PQ+PR is as small as possible.
    To solve this problem, we need to minimize the following function of x: f(x) = ?
    over the closed interval [A,B] where A = ? and B = ?
    $\displaystyle PQ = d_1$

    $\displaystyle d_1 = \sqrt{(x-0)^2 + (0-5)^2}$


    $\displaystyle PR = d_2$

    $\displaystyle d_2 = \sqrt{(x-10)^2 + (0-6)^2}$

    $\displaystyle d_1 + d_2 = S$

    find $\displaystyle \frac{dS}{dx}$ and minimize
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  3. #3
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    Still not getting it...

    Thank you for your help, I really appreciate it...

    But I am still not getting what the technique is here to answer the question.

    I understand finding distance (d1) and distance (d2)

    but the derivative part has me thrown off. Also, I don't get what function is supposed to be minimized...

    How do I go step by step through this?
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  4. #4
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    Quote Originally Posted by derekjonathon View Post
    Thank you for your help, I really appreciate it...

    But I am still not getting what the technique is here to answer the question.

    I understand finding distance (d1) and distance (d2)

    but the derivative part has me thrown off. Also, I don't get what function is supposed to be minimized...

    How do I go step by step through this?
    minimize the function $\displaystyle S = \sqrt{x^2+25} + \sqrt{x^2-20x+136}$

    start by finding $\displaystyle \frac{dS}{dx}$, set the result equal to 0, and solve for the value of x that minimizes S.
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