The Area of a
Bow Tie
Consider the
following bow-tie with the following dimensions:
A = length of
the large (outer) line
B = length of
the smaller (inner) line
H = half of
the height (or length) of the bow tie
The bow tie
is two trapezoids stacked on each other, with their narrow ends meeting each
other. Hence, the area of the bow tie is
twice the area of a trapezoid.
The area of a
trapezoid is:
A = 2 *(area
of the trapezoid)
A = 2 *(1/2 *
H * (A + B))
A = H * (A +
B)
Fairly
easy.
The Volume of
a Cylindrical Hourglass
I do not
think there is an official name for this, but I am naming it the cylindrical hourglass
because it is built by two identical cylindrical frustums, with their smaller
bases attached. Hence the volume of the hourglass is twice the volume of one of
the cylindrical frustum (frustum of a cylinder).
Let: a = A/2 = small (inner) radius,
b = B/2 =
larger (outer) radius, and
h = H =
height of half of the hourglass
We know the
volume of a frustum of a cylinder is:
V = h/3 * π *
(a^2 + a*b + b^2)
** For a
derivation of the volume of frustum of a cylinder, see the next section.
Since the
volume of the cylindrical hourglass is twice this volume, the final volume is:
V = 2 * h/3 *
Ï€ * (a^2 + a*b + b^2)
Volume of a
Frustum of a Cylinder
Imagine the
cross section of the frustum placed on a Cartesian plane. Let the bottom of the base “rest” on the
x-axis as seen on the above picture. We
will calculate the volume of the frustum using the Method of Discs. For this calculation, let the discs revolve
around the y-axis. The general formula
for volume with the discs revolving around the y-axis is:
V = Ï€ * ∫ (r(y))^2
dy
The boundaries
used for this calculation are:
Top
Boundary: y = h
Bottom
Boundary: y = 0
Left boundary:
x = 0
Right
boundary: This is a line. x = y * (a – b)/h + b. See below.
Slope =
change in y/change in x = (h – 0)/(a – b) = h/(a – b)
Determining
the line using point (b,0) and the right boundary:
y – 0 = h/(a –
b) * (x – b)
y * (a - b)/h
= x – b
y * (a - b)/h
+ b = x
r(y) = right
boundary – left boundary
r(y) = y * (a
- b)/h + b – 0
r(y) = y * (a
- b)/h + b
Calculating
Volume:
h
V = Ï€ * ∫ (y * (a - b)/h + b)^2 dy
0
h
V = Ï€ * ∫ (a – b)^2/h^2 * y^2 + 2*b/h * (a – b) * y +
b^2 dy
0
h
V = π * [ ((a
– b)^2 * y^3)/(3 * h^2) + (b * y^2 * (a – b))/h + b^2 * y ]
0
V = π * h/3 *
((a – b)^2 + 3 * b * (a – b) + 3 * b^2)
V = h/3 * π *
(a^2 + a*b + b^2)
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That concludes this blog entry. The next blog entry I plan to do is calculating the volume of a parabolic hourglass. Thanks to all my followers and for the comments - appreciate as always.
Until next time,
Eddie
This blog is property of Edward Shore. 2014
