Source:
http://mathnexus.wwu.edu/Archive/problem/detail.asp?ID=138
Joe Flubb had contracted
to replace a 2-mile section of
oil pipeline in the
far north. The replacement needed to
be done during the
coldest part of a bitter cold winter
many miles north
of the Arctic Circle. The thermometer
dropped to 40 Celsius
degrees below zero on the day Joe
and his crew installed
the new section of pipeline, so
understandably he
was anxious to get the job completed
as quickly as possible.
According to the specifications,
the new 2-mile pipeline
was to be firmly
anchored to the ground at each end. The
specs also required
that expansion joints be placed at
appropriate points
along the new pipe to allow for expan-
sion of the pipe
when the temperature would go up. This
precaution seemed
quite unnecessary to Joe because the
metal he was using
in the pipeline had an expansion
coefficient of only
0.00005. This means that every time
the temperature increases
by 1 degree Celsius, each foot
of pipe grows by
0.00005 feet. That's only 6 hundredths
of one percent of
an inch per foot of pipe. Clearly such
a minute expansion
could be ignored, Joe decided.
To make a drawing
of the extended pipeline after the
temperature increases,
points A and B are 2 miles apart
and where pipe is
firmly anchored to the ground. Point
C in the midpoint
of AB. For ease of computation, assume
that as the pipeline
expands, the pipe lifts off the
 |
ground
to a point D directly above point C
i.e. DC is perpendicular
to AB), forming
straight segments
AD and BD. Thus, point D
is the highest point
of the pipe. |
Photo from PlanetWare.com
Summer finally arrives.
The temperature soars to 30
degrees Celsius.
The pipe will expand a bit. Try to
guess and determine:
How high is CD?
Could a mouse squeeze
under the pipe at C?
Could a sled dog
squeeze under?
A polar bear?
.
.
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Drawing
from tom on 8/21/2009 |
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Note: This
"favorite" problem was written by Boyd Henry,
a highly-respected
mathematics educator in Idaho. It is
reprinted here in
his honor.
Hint: Make
a guess then use the Pythagorean Theorem
(for right triangles)
to find the height. The distance
from A to C is exactly
5280 feet (one mile). The distance
from A to D is the
expanded length of one mile of pipe.
Your assignment is
to find how long AD is and then find
the height of CD.
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Finger pointing down
from darrell94590 on 1/2/2006 |
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..
THE SOLUTION
Solution Commentary:
For
each degree of temperature
increase, the pipe
increases in length by a factor of 1.00005.
The 70 degree increase,
therefore, will for all practical
purposes increase
the pipe by a factor of 1.0035. Thus,
(5280 feet)x(expansion
factor of 1.0035) tells us that the
segment AD will expand
to 5298.48 feet. Using the Pythagorean
Theorem, we know:
CD2 = 5298.482 - 52802.
Upon solving for CD,
we discover that the pipeline will
raise an astounding
442 feet into the air. That is the height
of a 44-story building.
Does the answer change
if segments AD and DB are not assumed
to be straight, i.e.
the pipeline rises in an arc. First, is
the arc circular
or a segment of some other curve such as a
cantenary?
Second, what is the new length of DC? BEWARE:
These questions are
not trivial! For help, see discussion at
Ask
Dr. Math on the MathForem.
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Drawing
from tom on 8/21/2009 |
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.
Circular Arc Solution:
Assume
arc(AB) is part of a
circle where c =
2 pi r.
Lets compute the new
length of arc(AB) when the temperature
raises to 30°.
Let x represent the increase in arc(AB).
x = (70°)(0.0000.5)(5,280
feet/mile)(2 miles) = 36.96 feet.
Then the new circumference
= c + 36.96 = 2 pi R where
R is the new radius
of the circle.
Take the equation
c + 36.96 = 2 pi R and subtract the
original equation
c = 2 pi r from the first equation.
The new equation is
36.96 = 2 pi R - 2 pi r
Dividing everwhere
by 2 pi yields
36.96/(2 pi) = R - r
5.88 feet = R - r
R - r is the amount
the pipeline would raise which is 5.88 feet.
Both analysis appear
correct to me, but vary dramatically. |
