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The Hole Problem
Submitted by Richard Fisher, 03/15/1997.
Original answer by Valerio De Angelis;
this article by Allen Stenger.
A hole 6 inches long is drilled through a sphere of radius
R
to form a ring.
(See the figure below; the ring is 6 inches high and
10 inches across, and
R
is 5 inches.)
Find an expression for the volume of the ring. Is there anything remarkable
about this result?
Hint 1
The ring is a volume of revolution.
Hint 2
You can find the volume by the shell method or by the disk method,
but the disk method is easier. Turn the figure on its side, so we
are rotating about the x axis.
The ring is formed by rotating the
area bounded by the semi-circle
x^2 + y^2 = R^2,
y \ge 0
and the line of width 6 that intersects it, about the
x axis.
Let's write
y_c
for the y value of the circle, and
y_l
for the y value of the straight line.
Then the volume of the ring is
\pi \int_{-3}^3 (y_c^2 - y_l^2) \, dx
= \pi \int_{-3}^3 (R^2 - x^2) - (R^2 - 9) \, dx
= \pi \int_{-3}^3 (9 - x^2) \, dx
= 36 \pi \mathrm{\ in^3}.
Now, what is surprising about this result?
Click here for rest of the solution.
The Rest of the Solution
The volume is independent of
R;
no matter how large the sphere is, the ring has the same volume.
For very large spheres, the ring is very large in diameter but
the material is very thin, and these two factors
exactly compensate for each other, keeping the volume constant.
Is this a curiosity--a unique situation?
No. There are many figures of revolution, defined by the intersection
of two conic sections, that have similar properties; the volume is
independent of some of the parameters defining the curves. In our example
the conic sections are a circle and a line, and the volume depends only
on the height (in our drawing, the x extent)
of the intersection and not
on where the curves are placed. The paper by Alexanderson and Kosinski
cited below gives a table of many combinations with this property.
References
- Morris Kline,
Calculus: An Intuitive and Physical Approach,
Dover reprint, 1998. Exercise 5 on p. 447 states the problem in the
form we used here, asking you
"Is the answer remarkable in any respect?"
- G. L. Alexanderson and L. F. Kosinski,
"Some Surprising Volumes of Revolution",
Two Year College Math Journal,
v. 6 no. 3 (1975) pp. 13-15.
Reprinted on pp. 321-323 of:
Tom M. Apostol et al., editors,
A Century of Calculus, Part II,
Mathematical Association of America, 1992.
- Wikipedia has an article on the
Napkin-Ring Problem.
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