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Volume of a Conical Wedge


Date: 03/04/2002 at 17:14:12
From: Roger Staiger
Subject: A frustum problem 

The problem is this:

  Two porters agree to drink off a quart of strong beer
  between them, at two pulls, or a draught each, now the first 
  having given it a black eye, as it is called, or drank till 
  the surface of the liquor touched the opposite edge or the 
  bottom, gave the remaining part of it to the other.  What 
  was the difference of their shares, supposing the pote was 
  a frustum of a cone, the depth being 5.7 inches, the diameter 
  of the top 3.7 inches. and that of the bottom 4.23 inches?

The answer I've been given is 7.05 cubic inches, arrived at by the 
formula

  (D^(3/2) - d^(3/2))^2 / (D-d) * (pi h/3) 
 
where 

  D = 4.23
  d = 3.7
  h = 5.7

I approached this problem with the formula for the a frustum of a
cone.  But apparently, I have made some logic error in switching 
between volume of the frustum and the area of trapezoids. But I 
cannot replicate this formula.  

You assistance is appreciated.  Thanks,

Roger Staiger,    
Scotland MD


Hi, Roger.

I don't recall ever seeing a formula for the volume of a "conical
wedge" like this (or a name for it, for that matter); and I can't
find one anywhere either. Instead, I had to derive a formula for the
volume of the part of a cone left when part is cut away at an angle:

         pi h   R1^(3/2) [R1^(3/2) - R2^(3/2)]
    V1 = ---- * ------------------------------
           3               R1 - R2

The remaining part of the frustum of height h can be found just by
reversing R1 and R2, turning the frustum upside-down:

         pi h   R2^(3/2) [R1^(3/2) - R2^(3/2)]
    V2 = ---- * ------------------------------
           3               R1 - R2

The difference V1 - V2 is what we want, and subtracting these formulas
will indeed give the formula

           pi h   (R1^(3/2) - R2^(3/2))^2
    Vdif = ---- * -----------------------
             3            R1 - R2

Let's look at how I derived the formula for the volume of a "conical 
wedge."

First, I approached it in terms of an oblique slice through a complete 
cone:

                        + ------------------------
                       /|\\  d                  ^
                      / | \  \                  |
                     /  |  \    +               |
                    /   |   \  /                |
                   /    |    +                  |
                  /     |  / |\                 |
                 /      | /  | \                |
                /       +    |  \               |
               /      / |    |   \              |H
              /      /  |    |    \             |
             /     /    |    |     \            |
          2r/----+------+----|------\           |
           /    /|      |    |h      \          |
          /   /  |      |    |        \         |
         /  a/  h|      |    |         \        |
        /  /    -|      |    |          \       |
       / /      2|      |    |           \      |
      / /        |      |    |            \     |
     //          |  x/2 |    |       x     \    v
    +------------+------+----+--------------+ ----
    |         R         |          R        |

We know R and H, the radius and height of the full cone, and h, the
height of the cut. I'm going to find the volume of the top part of the
cone above the slice, by seeing it as an oblique elliptical cone. That
is, its base is the ellipse cut from the cone, and its height is d,
the perpendicular distance from this base to the apex. I have the find
the semiaxes a and b of the ellipse, and the height d.

Similar triangles give me the horizontal position of the end of the 
cut:

    x/h = R/H

    x = Rh/H

and the radius of the cone at the level of the center of the cut:

    r/R = (H-h/2)/H

    r = R(2H-h)/(2H)

The Pythagorean theorem gives

    a^2 = (R - x/2)^2 + (h/2)^2 = [(2R-x)^2 + h^2]/4

where a is half the length of the cut. (We won't actually need to have
done this!) To find b, half the width of the cut, consider a
horizontal cross-section through the cone at the center of the cut:

                    *********
                 **+         ***
                *  |\           *
               *   | \r          *
              *   b|  \           *
             *     |x/2\           *
             *-----+----+----------*
             *     |   /           *
              *   b|  /           *
               *   | /r          *
                *  |/           *
                 **+         ***
                    *********

Pythagoras gives

    b^2 = r^2 - (x/2)^2

Plugging in the values of r and x,

    b^2 = R^2(2H-h)^2/(2H)^2 - R^2h^2/(2H)^2

        = R^2/(4H^2) (4H^2 - 4Hh + h^2 - h^2)

        = R^2/(4H^2) ((2H-h)^2 - h^2)

        = R^2(H-h)/H

Now we have to find d. I'll redo the drawing with some different
lengths labeled for clarity:

                        +
                       /|\\  d
                      / | \  \
                     /  |  \    +
                    /  q|   \  /
                   /    |    +
                  /     |  / |\
                 /      | /  | \
                /       +    |  \
               /      / |    |   \
              /      /  |    |    \
             /     /    |    |     \
            /  t /      |    |      \
           /    /       |    |h      \
          /   /        p|    |        \
         /   /          |    |         \
        /  /            |    |          \
       / /              |    |           \
      / /               |    |            \
     //                 |    |      x      \
    +-------------------+----+--------------+
    |         R         |          R        |

Similar triangles give

    p/R = h/(2R-x)

so

    p = RH/(2R-x) 

      = RH/(2R - Rh/H) 

      = H^2/(2H-h)

and

    q = H-p = H - H^2/(2H-h) 

      = (2H^2 - Hh - H^2)/(2H-h) 

      = 2H(H-h)/(2H-h)

Again, by similar triangles, remembering that 2a is the length of the 
cut,

  t/R = 2a/(2R-x)

so

    t = 2Ra/(2R-x) 

      = 2Ra/(2R - Rh/H) 

      = 2aH/(2H-h)

Finally, similar triangles give

  d/q = R/t

so

    d = q*R/t 

      = R(H-h)/a

Now we can find the volume of our oblique elliptical cone:

   Vt = 1/3 Bh 

      = pi/3 abd 

      = pi/3 * b * ad

      = pi/3 * R sqrt[(H-h)/H] * R(H-h)

               R^2 * (H-h)^(3/2)
      = pi/3 * -----------------
                    H^(1/2)

Now we'd like to get this in terms of R1, R2, and h, where R1 is the
bottom radius, R2 is the radius of the top of the frustum containing
the wedge, and h is the height of that frustum. We have to find H
given R2:

                        + -----------------------
                       /|\                     ^
                      / | \                    |
                     /  |  \                   |
                    /R2 | R2\                  |
                   +----+----+                 |
                  /     |     \                |
                 /      |      \               |
                /       |       \              |
               /        |        \             |H
              /         |         \            |
             /          |          \           |
            /          h|           \          |
           /            |            \         |
          /             |             \        |
         /              |              \       |
        /               |               \      |
       /                |                \     |
      /                 |                 \    |
     /                  |                  \   v
    +-------------------+-------------------+ ---
    |         R1        |         R1        |

Similar triangles give

     R2/(H-h) = R1/H

    R1H - R1h = R2H

            H = R1h/(R1-R2)

          H-h = R2h/(R1-R2)

Plugging these into our volume,

                R1^2 * (H-h)^(3/2)
    Vt = pi/3 * ------------------
                     H^(1/2)

                R1^2 * R2^(3/2) h^(3/2)    (R1-R2)^(1/2)
       = pi/3 * ----------------------- * ----------------
                    (R1-R2)^(3/2)         R1^(1/2) h^(1/2)

                R1^(3/2) R2^(3/2) h
       = pi/3 * -------------------
                       R1-R2

Now we can find the volume of the bottom part, the actual wedge we are
looking for, by subtracting the top part from the whole cone:

    Vc = pi/3 R1^2 H

       = pi/3 R1^3 h / (R1-R2)

    Vb = Vc - Vt

                R1^3 h          R1^(3/2) R2^(3/2) h
       = pi/3 * ------ - pi/3 * -------------------
                 R1-R2                 R1-R2

                R1^(3/2) h [R1^(3/2) - R2^(3/2)]
       = pi/3 * --------------------------------
                             R1-R2

I then found the volume of the top part of the frustum by swapping R1
and R2 (you could also subtract Vb from the volume of the frustum),
and then found the difference.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
College Higher-Dimensional Geometry
High School Higher-Dimensional Geometry

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