Results 1 to 3 of 3

Math Help - area & volume of an open box

  1. #1
    Newbie
    Joined
    Oct 2006
    Posts
    18

    Post area & volume of an open box

    A carpenter has been asked to build an open box with a square base. The sides of the box will cost $3 per square meter and the base will cost $4 per square meter. What are the dimensins of the box of greatest volume that can be constructed for $48?

    Should I start with finding the Surface Area or the Volume?
    I'm not sure if I remember the formulas:
    SA = 2b^2 + 4bh
    V = b^2h
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by confusedagain View Post
    A carpenter has been asked to build an open box with a square base. The sides of the box will cost $3 per square meter and the base will cost $4 per square meter. What are the dimensins of the box of greatest volume that can be constructed for $48?

    Should I start with finding the Surface Area or the Volume?
    I'm not sure if I remember the formulas:
    SA = 2b^2 + 4bh
    V = b^2h
    Let the side of the box be b, and the height be h, then the volume is:

    V = b^2 h,

    and the cost is:

    C = 4 (b^2) + 3 (4 b h) = 4 b^2 +12 b h = 48

    divide through by 4:

    b^2 + 3 b h = 12.

    So:

    h = (12 - b^2)/(3 b)

    Hence:

    V = b^2 (12 - b^2)/(3 b) = b (12 - b^2)/3 = 4 b - b^3/3

    The maximum volule will occur when dV/db=0, or:

    dV/db = 4 - b^2 = 0

    which has roots b=+/-2. The second derivative test tells us that b=+2
    gives a maxima (that is d^2V/db^2<0 at b=2).

    So b=2, then as h = (12 - b^2)/(3 b), h = 4/3

    RonL
    Last edited by CaptainBlack; March 13th 2007 at 10:58 AM. Reason: Incorporate Soroban's correction
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,546
    Thanks
    539
    Hello, confusedagain!

    The Captain made a teensy error . . .


    Quote Originally Posted by CaptainBlack

    V .= .b(12 - b)/(3b) .= .b(12 - b)/3 .= .4b - b/3

    The maximum volume will occur when dV/db = 0, or:
    . . dV/db .= .4 - b .= .0
    Therefore: .b = 2, h = 4/3

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Find the surface area of an open-topped box
    Posted in the Geometry Forum
    Replies: 2
    Last Post: November 15th 2011, 01:50 PM
  2. [SOLVED] Modeling Volume of an Open Box
    Posted in the Geometry Forum
    Replies: 2
    Last Post: March 9th 2010, 07:57 PM
  3. An open box......volume maximzed
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 18th 2009, 07:03 PM
  4. Replies: 1
    Last Post: May 12th 2009, 12:36 AM
  5. finding the volume and the domain of an open box
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: February 14th 2009, 05:51 AM

Search Tags


/mathhelpforum @mathhelpforum