Pipe Traversal Through a Cornered Hallway
Robert Gubler
The purpose of this project is to determine the maximum pipe length that will fit around a hallway defined by the constant width of passage a, the constant width of passage b, and the constant height of the hallway c, where passage a and b meet at exactly 90○. In addition, the problem has been simplified such that the width of the pipe is negligible. The project presents two circumstances; the first where lead pipe is used and must be slid horizontally along the floor, and the second where titanium pipe is used and may be lifted vertically. Determining the maximum pipe length that will fit around the hallway will allow us to derive the total number of pipe segments, and joints, needed to move p feet of plumbing through the hallway. In turn we will able to develop generic equations for determining the total cost of the job for either circumstance; assuming lead pipe costs l dollars per foot, titanium costs t dollars per foot and joints cost j dollars each.
Rotating the pipe around corner B will allow us to determine the longest possible pipe length that will successfully make it through the hallway. Assuming the pipe, represented by the line ABC, is always resting against the hallway walls the length of the pipe will at first decrease as it is rotated around the corner, then finally increase again. The maximum length of pipe that will fit around the corner can be found be determining the shortest line between point A and C, when A and C are always resting against the walls.
Therefore when:
d1
= 
=
= 
d2
= 
= 
= 
From this we can see that the total length of the pipe is:
Llp
= 
Taking the derivative of Llp
with respect toq will allow
us to find the local minimum of Llp in the domain
. From this we will be
able to determine the maximum length a pipe can be and still make it through
the hallway.
To do this we first differentiate Llp:
Llp¢= 
Then to find the local minimum in domain of 
we solve Llp’for zero:
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0 |
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q |
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Therefore, when we plug the critical point,
, into the function of Llp we can find the maximum length a pipe can be, for any given a and b, and still while being slid
across the floor make it around the corner of the hallway. This equation can be further simplified by
using the Pythagorean Theorem, y2
+ x2 = r2.
By massaging the equation that gives us the critical point we can separate two sides of the Pythagorean Triangle into:
, and
, and find r to be
Following from this we can see that two angles of the Pythagorean Triangle are:
and
Since Llp = = 
we can replace the
reciprocals
and
with their respective expressions referenced above. Doing this will yield the following equation
for Llp:
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Llp |
= |
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Finally we can conclude that the simplified equation for finding the maximum length of a pipe that can successfully be slid around a corner of a hallway where the widths of the passage walls are defined by any given a and b is:
Llp = 
In the second scenario titanium pipe is used, and may be
lifted, vertically at an angle. To
determine the maximum length a pipe can be when lifted at an angle we must take
into account the height of the hallway.
From the diagram below we can see that the opposite side of the triangle relates directly to the maximum
length a pipe can be if it were slid
around the corner. We can use both the
equation obtained for Llp and the height c
in the Pythagorean Theorem to find the length of the hypotenuse side of the triangle.
This is the maximum length a pipe can be when lifted at an angle and still make it around the
corner of the hallway.
When:
x = 
y = c, the height of the hallway
Then:
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Lti |
= |
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Therefore, the maximum length a pipe can be when lifted at an angle is:
Lti =
As a result of finding the formulas and we can determine the number of joints needed between pipe segments, their cost, and the total cost of the job.
Number of Joints required when using lead pipe, where P is the total length of pipe needed:
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Total Cost of joints when using lead pipe, where J is the total number of joints:
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Total cost of entire Job when using lead pipe, where l is the cost of lead pipe per foot:
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Number of Joints required when using titanium pipe, where P is the total length of pipe needed:
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Total Cost of joints when using titanium pipe, where J is the total number of joints:
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Total cost of entire Job when using titanium pipe, where t
is the cost of lead pipe per foot:
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