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 Subj:.....The Alaskan Pipeline (S668)           From: MathNexus.wwu.edu           on 11/2/2008 Drawing from MathNexus...

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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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 Drawing from Ripleys-Believe It Or No
..

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.

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