Results 1 to 3 of 3

Math Help - Optimizing Area-HELP!

  1. #1
    Member
    Joined
    Jul 2007
    Posts
    75

    Thumbs down Optimizing Area-HELP!

    Your parents are going to knock out the bottom of the entire length of the south wall of their house and turn it into a green house by replacing some bottom portion of the wall by a huge sloped piece of glass (which is expensive). They have already decided they are going to spend a certain fixed amount. The triangular ends of the greenhouse will be made of various materials they already have lying around.
    The floor space in the greenhouse is only considered usable if they can both stand up in it, so part of it will be unusable, but they don't know how much. Of course this depends on how they configure the greenhouse. They want to choose the dimensions of the greenhouse to get the most usable floor space in it, but they are at a real loss to know what the dimensions should be and how much usable space they will get.

    I realy need help with this! I don't even know where to begin. Please help me!
    Last edited by mr fantastic; November 18th 2009 at 02:47 AM. Reason: Removed excessive !'s in post title and post
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2008
    Posts
    1,034
    Thanks
    49
    This is a bit like all those ladder-to-wall-across-a-fence problems, but you're minimising something other than the ladder.

    Start by relabelling the nice picture at http://www.mathhelpforum.com/math-he...dder-wall.html. Let's use capitals for constants.

    Z = length of the ladder / glass roof

    H = height of the fence / tallest person.

    x = distance of roof-end from wall

    y = height of other roof-end up wall, but

    y = \sqrt{Z^2 - x^2}

    and finally, u = usable floor.

    Use similar triangles to see that the ratio of (x - u) to H is the same as that of x to y.

    Express u as a function of x, with constants Z and H. Differentiate u with respect to x and set to zero to find the best x.

    Just in case a picture helps differentiate...



    ... where



    ... is the chain rule, here wrapped inside...



    ... the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

    In the bottom row, the fourth balloon along (second from left in the big network) is unchanged on its way down from the top row (it wasn't its turn - by the product rule - to get differentiated) but I've tweaked it for the sake of a common denominator.

    Then set the derivative to zero. You can thus offer your parents an equation involving x = the optimum extension from the wall, and H and Z for which (presumably) they know the values. I doubt whether we can solve algebraically for x in terms of H and Z... but I'm not at all sure.

    __________________________________________

    Don't integrate - balloontegrate!

    Balloon Calculus: Gallery

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Last edited by tom@ballooncalculus; November 18th 2009 at 07:14 AM. Reason: corrected mistake
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by iheartthemusic29 View Post
    Your parents are going to knock out the bottom of the entire length of the south wall of their house and turn it into a green house by replacing some bottom portion of the wall by a huge sloped piece of glass (which is expensive). They have already decided they are going to spend a certain fixed amount. The triangular ends of the greenhouse will be made of various materials they already have lying around.
    The floor space in the greenhouse is only considered usable if they can both stand up in it, so part of it will be unusable, but they don't know how much. Of course this depends on how they configure the greenhouse. They want to choose the dimensions of the greenhouse to get the most usable floor space in it, but they are at a real loss to know what the dimensions should be and how much usable space they will get.

    I realy need help with this! I don't even know where to begin. Please help me!
    1. Let G denote the length of the glas pane (because the price for the glas is fixed G is a constant), P the length of the longest person, b the part of the wall which has to be taken off, a the complete floor and x the usable part of the floor. (see attachment)

    2. Define a coordinate system with the floor on the x-axis and the wall on the y-axis. G is a segment of the line

    y=-\frac ba x+b

    3. According to Pythagorean theorem you get:

    b^2+a^2 = G^2~\implies~a = \pm \sqrt{G^2-b^2}
    That means the equation of the line becomes:

    y=-\frac b{\sqrt{G^2-b^2}} x+b (Since a is pointing to the right I took only the positive value of a)

    4. Now y = P (which is for a certain period of time a constant)

    P=-\frac b{\sqrt{G^2-b^2}} x+b~\implies~x=\dfrac{(b-P) \sqrt{G^2-b^2}}{b}

    This is an equation of the function x(b). Differentiate x wrt b and solve for b the equation x'(b) = 0.

    5. I've got:

    x'(b)= \dfrac{G^2 \cdot P-b^3}{b^2 \sqrt{G^2-b^2}}

    and

    b = \sqrt[3]{G^2 \cdot P}

    6. Plug in this value into x(b) - and you'll get a really nasty looking term for x.
    Attached Thumbnails Attached Thumbnails Optimizing Area-HELP!-gewaechshausanwand.png  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Optimizing time
    Posted in the Calculus Forum
    Replies: 10
    Last Post: April 4th 2011, 12:55 PM
  2. Optimizing
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 8th 2010, 11:16 AM
  3. Optimizing
    Posted in the Calculus Forum
    Replies: 0
    Last Post: November 1st 2009, 01:50 PM
  4. Optimizing Distance
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 20th 2009, 04:45 PM
  5. Optimizing a rectangle
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 18th 2009, 05:55 PM

Search Tags


/mathhelpforum @mathhelpforum