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## An Expanding Experience

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 expansion 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 thta every time the temperature increases by 1 degee Celsius, each foot of pipe grows by 0.00005 feet. That's only 6 hundreths 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 perpedicular to AB), forming straight segments AD and BD. Thus, point D is the highest point of the pipe.

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?
NOTE: This "favorite" problem was written by Boyd Henry, a highly-respected mathematics educator in Idaho. It is reprinted here in his honor.

Source: Washington Mathematics, Spring 1991, p. 7

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.

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 ar 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.