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Dimensional analysis and frustum of a cone

  1. May 9, 2008 #1
    1. The problem statement, all variables and given/known data
    Hi
    Im having some difficulty with the following question:
    Figure P1.14 shows a frustrum of a cone. Of the following mensuration (geometrical) expressions, which describes (a) the total circumference of the flat circular faces, (b) the volume, and (c) the area of the curved surface?
    (i) π(r1 + r2)[h2 + (r1 – r2)2]1/2 (ii) 2π(r1 + r2) (iii) πh(r1^2 + r1r2 + r2^2)

    and Im at part b.
    Since I already know that the volume of a frustum of a cone is number (iii) I have to now prove it.
    The problem is that Im having some difficulty showing it with the use of dimensional analysis.

    Since V=[L^3]
    how is possible that πh(r^12 + r1r2 + r2^2) is equal to it?
    I know that r=L and h=L but completly confused on how to set it up
    can someone point me in the right direction? I dont


    thank you
     
  2. jcsd
  3. May 10, 2008 #2
    First start off with a cone, say cone A. If you slice off a smaller cone, say cone B, from the "top" of cone A, the solid that remains is a frustrum.

    For this question, I believe [tex]r_1[/tex] is the radius of the circular base of cone A (equivalently, the larger flat circular face of the frustrum) and [tex]r_2[/tex] is the radius of the circular base of cone B (equivalently, the smaller flat circular face of the frustrum). I think the quantity h refers to the height of the frustrum.

    Here are some useful hints for solving part (b) of the question:
    1) How do you find the volume of a cone?
    2) Cones A and B are similar; use this to express the heights of cones A and B in terms of [tex]r_1[/tex], [tex]r_2[/tex] and h.
    3) Note that [tex]x^3 - y^3 \ = \ (x-y)(x^2+xy+y^2)[/tex].

    To clarify some of the expressions in your original post,
    Option (i) [tex]\pi(r_1+r_2)\sqrt{h^2+(r_1-r_2)^2}[/tex]
    Option (iii) [tex]\frac{1}{3}\pi h (r_1^2+r_{1}r_{2}+r_2^2)[/tex]
     
    Last edited: May 10, 2008
  4. May 11, 2008 #3
    thank you, I think I got it
     
  5. Aug 26, 2010 #4
    I'm struggling on practically this exact problem right now in my textbook.

    The way I see it, I get this:

    L (L2 + L2 + L2) = L3 + L3 + L3 = 3L3

    But I don't see how I can arrive at L3 from my end answer...
     
  6. Feb 28, 2012 #5
    I just had to answer my own question here. The 3 disappears because it's dimensionless!
     
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