Results 1 to 4 of 4

Thread: Minimum Area

  1. #1
    Member McScruffy's Avatar
    Joined
    Jul 2009
    Posts
    87
    Awards
    1

    Minimum Area

    Need some help with this one.

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

    I'm having some trouble with the setup. Any help would be appreciated.
    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member malaygoel's Avatar
    Joined
    May 2006
    From
    India
    Posts
    648
    Quote Originally Posted by McScruffy View Post
    Need some help with this one.

    $\displaystyle \overline {PQ} $ and $\displaystyle \overline {SR} $ intersect two parallel lines, and each other at a point $\displaystyle T$ on the interior of the parallel lines. Point $\displaystyle R $ is $\displaystyle d $ units from $\displaystyle P $. How far from $\displaystyle Q $ should the point $\displaystyle S $ be so that the sum of the triangles $\displaystyle \triangle STQ $ and $\displaystyle \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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member McScruffy's Avatar
    Joined
    Jul 2009
    Posts
    87
    Awards
    1
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member malaygoel's Avatar
    Joined
    May 2006
    From
    India
    Posts
    648
    Let,

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

    n be the distance between parallel lines

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

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

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

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

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

    Substitute value of $\displaystyle h_1$ and $\displaystyle h_2$ in equation 1 and differentiate with respect to m to maximize area.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Minimum area of cylinder
    Posted in the Geometry Forum
    Replies: 2
    Last Post: Mar 6th 2011, 05:31 AM
  2. Finding minimum area
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Jan 8th 2011, 06:22 PM
  3. Help with Minimum Area
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Mar 27th 2009, 01:24 AM
  4. Minimum total area
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Feb 21st 2009, 03:30 PM
  5. Minimum Area Question
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Dec 16th 2008, 09:54 PM

Search Tags


/mathhelpforum @mathhelpforum