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Optimization, Minima, Open Top Box

  1. Jan 8, 2007 #1
    Optimization, Minima, new question:Sheet Alluminum

    1. The problem statement, all variables and given/known data
    A box with a square base and no top must haave a volume of 10000 cm^3. If the smallest dimension in any direction is 5 cm, then determine the dimensions of the box that minimize the amount of material used.


    2. Relevant equations
    Volume: x^2y
    Surface Area: x^2+4xy



    3. The attempt at a solution
    Let x represent the length and width of the box
    Let y represent height

    x>5

    I drew a neat little drawing of the box and labeled it according to the statements above.

    V=x^2y
    10000=x^2y
    y=10000/x^2


    SA=x^2+4xy
    =x^2+4x(10000/x^2)

    Now I think that I need to set the SA equation to zero then differentiate but I can't quite remember what I'm doing with it. I'd appreciate anyhelp that could be offered.


    ~Thanks!
     
    Last edited: Jan 9, 2007
  2. jcsd
  3. Jan 8, 2007 #2
    Don't forget to subtract the area of the top from the surface area equation.
     
  4. Jan 8, 2007 #3

    Gib Z

    User Avatar
    Homework Helper

    He never added it lol.

    Your on the right track. [tex]SA=x^2+\frac{40000}{x}[/tex]

    Differentiate with respect to x and set equal to zero. you get [tex]2x=\frac{40000}{x^2}[/tex] Take the x^2 over and we get 2x^3=40000, x^3=20000. Take the cube root, we get around 27.144cm. Sub that value Into the 10000=x^2 y to get y.
     
  5. Jan 9, 2007 #4
    wow.. Thanks that make sense I think, been a couple months since we did this in class.

    Thanks
    ~RS

    So then x=27.144 cm and y=6.786 cm.
    Don't I also sub 5 from x>5 into the equation to tie up loose ends and make sure that it really is 27 cm? oops wrong equation y=13 and change.
     
    Last edited: Jan 9, 2007
  6. Jan 9, 2007 #5
    A cylindrical can is to hold 500 cm^3 of apple juice the design must take into account that the height must be between 6 and 15 cm, inclusive. How should the can be constructed so that a minimum amount of material will be used in the construction? assume no waste.

    V is volume and SA is surface area

    so far this is what I have:

    15>h>6

    v=(pi)r^2h
    500=(pi)r^2h
    500/(pi)r^2=h

    SA=2(pi)r(500/(pi)r^2) + 2(pi)r^2

    Now I believe I need to find the derivative and set it to zero then solve for r.

    The problem is that I can't quite figure out how I'm supposed to find the derivative if someone could give me a hand understanding the process as I have forgotten and my teacher is busy with the latest stuff.
     
  7. Jan 9, 2007 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    No, your requirement is that x and y must both be greater than 5 which is true.
     
  8. Jan 9, 2007 #7

    Sorry I forgot to say that I finally got that question finished and it seems to be correct, I had been struggling with it for the last few days and my teacher wasn't able to explain it in a way that helped. Thanks to Gib Z it clicked and I figured it out as far as boxes go, but when I see derivatives and pi I get thrown off.

    Thanks,
     
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