Calculus Pipeline Project
Curtis Casados
July 10, 2014
Professor David Kuralt
Project Scope:
The U.S. Interior Secretary recently approved drilling of natural gas wells near Vernal, Utah. Your company has begun drilling and established a highproducing well on BLM ground. They now need to build a pipeline to get the natural gas to their refinery.
While running the line directly to the refinery will be the least amount of pipe and shortest distance, it would require running the line across private ground and paying a rightofway fee. There is a mountain directly east of the well that must be drilled through in order to run the pipeline due east. Your company can build the pipeline around the private ground by going 5 mile directly west and then 15 miles south and finally 40 miles east to the refinery (see figure below). Cost for materials, labor and fees to run the pipeline across BLM ground is $500,000 per mile.
Cost of drilling through the existing mountain would be a onetime cost of $2,000,000 on top of the normal costs of the pipeline itself. Also the BLM will require an environmental impact study before allowing you to drill through the mountain. Cost for the study is estimated to be $320,000 and will delay the project by 4 months costing the company another $120,000 per month.
For any pipeline run across private ground, your company incurs an additional $350,000 per mile cost for rightofway fees.
Your company has asked you to do the following:
A. Determine the cost of running the pipeline strictly on BLM ground with two different cases:
a. One running west, south and then east to the refinery
otal Distance 5 miles west 15 miles south 40 miles east 60 milesT = + + = otal Cost 0 miles ($500, 00 per/mile) $30, 00, 00.00T = 6 0 = 0 0
b. One heading east through the mountain and then south to the refinery
otal Distance 35 miles east 15 miles south T = + ost per mile $500, 00.00 C = 0 rilling $2, 00, 00.00 D = 0 0 nvironmental Impact $320, 00.00 E = 0 elay Cost $480, 00.00 D = 0
otal Cost 50 miles ($500, 00.00) 2, 00, 00.00 $320, 00.00 $480, 00.00 27, 00, 00.00 T = 0 + $ 0 0 + 0 + 0 = $ 8 0
B. Determine the cost of running the pipeline:
a. The shortest distance from the well to refinery across the private ground
iagonal Private Land Distance 38.07886553 miles D = √(15) 35)2 + ( 2 = rivate Land Cost $850, 00.00 per mile P = 0
otal Cost 38.07886553 miles ($850, 00.00) $32, 67, 35.70 T = 0 = 3 0
b. The shortest path across the private ground (directly south), then straight to the refinery (directly east).
rivate Land Distance 15 miles south P = LM Distance 5 miles east B = 3 rivate Land Cost $850, 00.00 per mile P = 0 LM Cost $500, 00.00 per mileB = 0
otal Cost 15 miles ($850, 00.00) 35 miles ($500, 00.00) $30, 50, 00.00 T = 0 + 0 = 2 0
C. Determine the optimal place to run the pipeline to minimize cost. Clearly show all work including drawing the pipeline on the figure below. Make it very clear how you use your knowledge of calculus to determine the optimal placement of the pipeline
ptimal Private Land Distance milesO = √15 2 + x2 LM Distance 35 miles miles)B = ( − x rivate Land Cost $850, 00 per mile P = 0 LM Cost $500, 00.00 per mileB = 0
ost Function c(x) 50, 00 00, 00(35 )C = 8 0 √152 + x2 + 5 0 − x
a. Take First Derivative
(x) /dx[850, 00(15 ) ] /dx[5, 0, 00(35 )] c′ = d 0 2+ x2 1/2 + d 0 0 − x
/dx(850, 00)(15 ) /dx(15 )(850, 00) /dx(500, 00)(35 ) /dx(35 )(500, 00) = d 0 2+ x2 1/2+ d 2+ x2 0 + d 0 − x + d − x 0
/2(15 ) /dx(15 )(850, 00) − )(500, 00) = 1 2+ x2 −1/2× d 2+ x2 0 +0+ ( 1 0
/2(15 ) x(850, 00) 00, 00 = 1 2+ x2 −1/2×2 0 −5 0
(850, 00) 00, 00= 2x2(15 +x )2 2 0 −5 0
b. Find Critical Numbers
00, 00x =
√15 +x2 2
850,000x − 5 0 = 0
00, 00x = 5 0 =
√15 +x2 2
850,000x
00, 00 50, 00xx = 5 0 √152 + x2 = 8 0
x = 850,000500,000√15 +x2 2
= x
x = (1017) 1710√15 +x2 2
= x (1017)
x = (√152 + x2)2 = x(1017 )2
5 x = 1 2 + x2 = x2 (100289) 25 x = 2 + x2 = x2 (100289) 25 x = 2 = x2 (100289) − x2 25 x = 2 = x2 (100289) − 1
x =√ 225289/100−1 = x
±x = 2150√21
0.91 Critical Numberx = 1
otal Cost c(10.91) 850, 00 00, 00(35 0.91) $27, 10, 95.33 T = = 0 (√225 0.91+ 1 2) + 5 0 − 1 = 8 7
D. Graph
E. Reflection Pipeline Project Reflection
In this reflection I will describe what the final project assignment was and then I will explain
how I used calculus to find a solution that I did not previously know existed or even could exist. After that I
will explain how calculus has already impacted my life and how I can see it being something very useful in
the future especially towards my career path.
So the pipeline project was the final project for the 1020 calculus course that is required and the
question that it wanted solved was, what was the cheapest way to lay a pipe for an oil company but the
catch was there were 3 obvious ways but as I mentioned before with the use of calculus there was
another way to lay the pipe. I don’t think I would have been able to figure out that there was another way
by just using trigonometry or at least not as efficiently. It ended up being the second scenario was the
cheapest route but it was still very interesting to learn that there was another route that was opened up
with the use of calculus.
I have learned many things so far in this class, I found limits to be very interesting not only because
they are the foundation for a lot of calculus one concepts but also because they can give you a numerical
approximation that is so close that it is eventually considered the limit. I was also very interested in
L’Hospital’s theorem. I think the reason I was so intrigued by L’Hospital was because I have been told all
of my life basically that you cannot divide by 0 or 0/0 doesn’t exist but L’Hospital proved that a limit does
exist if you just take the derivative of the top and the derivative of the bottom of the equation separately.
Now I will explain how calculus has already impacted my life and how I can see it being such a useful tool
in the future. I was talking with an old friend (who has taken calculus as well) about physics honors and we
both just started chuckling when we started to think about acceleration, velocity, and distance questions
from physics honors because now we see that we can just take the integral of them and come up with an
answer in about half of the time as it used to take. So I can see calculus having many useful applications
in the near future especially if I am to continue to pursue a degree in mechanical engineer.