Results 1 to 6 of 6

Thread: The dimensions of the largest-volume rectangular box

  1. #1
    Newbie
    Joined
    Jun 2017
    From
    USA
    Posts
    1

    Unhappy The dimensions of the largest-volume rectangular box

    What are the dimensions of the largest-volume rectangular box, with a square base and an open top , that can be made from 10 000 cm^2 of materials ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2010
    Posts
    2,834
    Thanks
    1087

    Re: The dimensions of the largest-volume rectangular box

    Here is the set up: we have a base and four sides. Each side has one dimension in common with the base and one dimension in common with each other (the height). Let's say that the base has dimensions $x\times x $ and the height is $y$. Then the surface area is $x^2+4xy$. You know that is equal to $10,000\text{ cm}^2$. You are trying to maximize volume: $V=x^2y$. So, solve for $y$ in the first equation, plug it into the volume formula, and then find the critical points for $x$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    16,107
    Thanks
    3647

    Re: The dimensions of the largest-volume rectangular box

    If this problem is being done on a precalculus level (i.e. , you haven't learned about derivatives), then get volume in terms of $x$ by solving $x^2+4xy = 10000$ for $y$, then sub the result into $V = x^2y$ to get volume strictly in terms of $x$.

    ... determine the maximum by graphing the single-variable volume equation on a calculator.

    If you do know to find critical values, then proceed with that method.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    1,671
    Thanks
    313

    Re: The dimensions of the largest-volume rectangular box

    Hmmm...since a square provides maximum area
    (a rectangle is a special square), wouldn't you
    simply "stick together" 5 squares, each with
    area=2000 sq.ft. ?

    Or should I get out of the boxing business?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    16,107
    Thanks
    3647

    Re: The dimensions of the largest-volume rectangular box

    Quote Originally Posted by DenisB View Post
    Hmmm...since a square provides maximum area
    (a rectangle is a special square), wouldn't you
    simply "stick together" 5 squares, each with
    area=2000 sq.ft. ?

    Or should I get out of the boxing business?
    ... your idea would work if the top were not open (6 square faces)
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    1,671
    Thanks
    313

    Re: The dimensions of the largest-volume rectangular box

    Yesss.....thanks Mike.

    x = 58, y = ~28.6; volume ~96222
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Apr 28th 2010, 08:01 PM
  2. Replies: 1
    Last Post: Apr 28th 2010, 07:55 PM
  3. Replies: 0
    Last Post: Feb 23rd 2010, 09:35 AM
  4. volume of largest rectangular box
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 1st 2009, 08:03 PM
  5. Dimensions of Rectangular Garden
    Posted in the Geometry Forum
    Replies: 2
    Last Post: Sep 19th 2008, 04:51 AM

Search Tags


/mathhelpforum @mathhelpforum