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Math Help - Minimum Area

  1. #1
    Member McScruffy's Avatar
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    Minimum Area

    Need some help with this one.

     \overline {PQ} and  \overline {SR} intersect two parallel lines, and each other at a point T on the interior of the parallel lines. Point  R is  d units from  P . How far from  Q should the point  S be so that the sum of the triangles  \triangle STQ and  \triangle PTR is a minimum?

    I'm having some trouble with the setup. Any help would be appreciated.
    Thanks.
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  2. #2
    Super Member malaygoel's Avatar
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    Quote Originally Posted by McScruffy View Post
    Need some help with this one.

     \overline {PQ} and  \overline {SR} intersect two parallel lines, and each other at a point T on the interior of the parallel lines. Point  R is  d units from  P . How far from  Q should the point  S be so that the sum of the triangles  \triangle STQ and  \triangle PTR is a minimum?

    I'm having some trouble with the setup. Any help would be appreciated.
    Thanks.
    Do you have some more information???

    However, given information says that:
    There are two parallel lines.
    P is a point on 1st parallel line, and Q on the other.
    R is on the same line as P, and S on the same line as Q.
    PQ and SR intersect at T.
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  3. #3
    Member McScruffy's Avatar
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    That is all of the information given in the problem. I can visualize the problem, the trouble that I'm having is getting an equation to differentiate.
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  4. #4
    Super Member malaygoel's Avatar
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    Let,

    m be the distance between S and Q (which we have to eventually determine)

    n be the distance between parallel lines

    h_1 be the altitude of \triangle PTR i.e. area of \triangle PTR= \frac{1}{2}dh_1

    h_2 be the altitude of \triangle STQ i.e. area of \triangle STQ= \frac{1}{2}mh_2

    hence, sum= \frac{1}{2}dh_1+\frac{1}{2}mh_2...........(1)

    Now,
    h_1=\frac{nd}{d+m}

    h_2=\frac{nm}{d+m}

    Substitute value of h_1 and h_2 in equation 1 and differentiate with respect to m to maximize area.
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