2014
Tina Owens
[CALCULUS PIPELINE PROJECT]
Pipeline project to minimize the cost of running gas pipelines from the natural gas well to the refinery.
The U.S Interior Secretary approved drilling of natural gas wells near
Vernal, Utah. My company will be doing the drilling and I have been asked to
determine the optimal place to run the pipeline to minimize cost.
Cost
 Materials, labor and fees to run the pipeline across BLM ground is
$500,000 per mile
 One-time fee to drill through the existing mountain of $2,000,000 +
normal costs
 Environmental impact study to allow for drilling would cost $320,000 +
4 month delay costing $120,000 per month
 Right-of-way fee to run across private land will be an additional
$350,000 per mile
Possible routes to the refinery
1) Running West, South and then East to the refinery strictly on BLM
land.
2) Running East through the mountain, then South to the refinery
strictly on BLM land.
3) Taking the shortest distance through private land to the refinery.
4) The shortest distance across private land (South), the East to the
refinery.
BLM Ground
BLM Ground
12 miles
4 mile
Private Ground
36 miles - BLM Ground
Route 1- Running West, South and the East to the refinery
BLM Ground
BLM Ground
12 miles
4 mile
Private Ground
36 miles - BLM Ground
Route 1 will avoid the cost of drilling the mountain, research fees, cost of
delay, and the right-of-way fees from drilling through private lane. However
route 1 is the longest way to the refinery, 52 miles.
52 miles × $500,000 per mile= $26,000,000.00 Total cost for route 1
Route 2- Running East through the mountain and then south to the refinery
BLM Ground
BLM Ground
12 miles
4 mile
Private Ground
36 miles - BLM Ground
Route 2 is 4 miles shorter than route 1 but it does include the cost of drilling
through the mountain as well as the cost for the environmental impact study
which will delay the project by 4 months.
44 miles × $500,000 per mile =22,000,000.00 cost of running pipe
+ 2,000,000.00  drilling fee
+ 320,000.00  study cost
+ 480,000.00  delay cost
Total cost = 24,800,000.00 for route 2
Route 3- The shortest distance from the well to the refinery across private
land
BLM Ground
BLM Ground
12 miles
4 mile
Private Ground
36 miles - BLM Ground
32 miles
12 miles
322 + 122 = 𝑐 2
C= √1168 ~34.18 miles
C
Cost
Right-of-way fee  $350,000 per mile
Cost to lay pipe  $500,000 per mile
Total cost per mile = $850,000.00 per mile
34.18 miles × $850,000.00 per mile = 29,053,000.00 Total cost for route 3
Route 3 is the most expensive.
Route 4- shortest path across private ground(South), then East to the
refinery.
BLM Ground
BLM Ground
12 miles
4 mile
Private Ground
36 miles - BLM Ground
Private land 12 miles × $850,000 per mile =$10,200,000.00
BLM land  32 miles × $500,000 per mile =$16,000,000.00
Total cost $10,200,000 + 16,000,000= 26,200,000.00
Cheapest Option
BLM Ground
BLM Ground
12 miles
4 mile
12
P
Private Ground
X
X-32
36 miles - BLM Ground
P=√122 + (32 − 𝑥)2
32 − 𝑥 =√𝑃2 − 122
So 𝑥 = 32 − √𝑃2 − 144
Cost= Private(P)+ BLM (𝑥)
Cost= 850,000𝑃 + 500,000(32 − √𝑃2 − 122 )
Cost
𝑑𝑐𝑜𝑠𝑡
𝑑𝑃
= 850,000 + 500,000 (−
0= 850,000 + 500,000 (−
𝑃
𝑃
)
√𝑃2 −144
)
√𝑃2 −144
500,000P=850,000√𝑃2 − 144
500,000P=850,000√𝑃2 − 144
500,00002 𝑃2 = 850,0002 𝑃2 − 850,0002 × 144
P=√
850,0002 ×144
850,0002 −500,00002
P= √
104040000000000
472500000000
P=√220.1905  P=14.84 miles
𝑥=32 − √14.842 − 144
𝑥=32-√76.2256
𝑥=32-8.731 𝑥=23.27 miles
Cost= 850,000(14.84) + 500,000(23.27)
Cost= 12,614,000+11,635,000 Cost= $24,249,000.00
Cost Function
27500000
27000000
26500000
Cost
26000000
25500000
25000000
24500000
24000000
23500000
23000000
Miles on Private Land
Minimum value at 14.84 miles on private land.
27
26.4
25.8
25.2
24.6
24
23.4
22.8
22.2
21.6
21
20.4
19.8
19.2
18.6
18
17.4
16.8
16.2
15.6
15
14.4
13.8
13.2
12
12.6
22500000
As shown by my calculation and by the graph of the cost function, the
most cost effective method is to go partially throughboth private land and
BLM ground. The pipe should go 14.84 miles through private land, costing
$12,614,000.00; the rest of the pipe will go through BLM ground for 23.27
miles, costing $11,635,000.00. The total cost for the project using the most
cost effective route will be $24,249,000.00.
Reflective Writing
Trying to find a use for calculus in my daily life has been a struggle. But
this project has shown me how calculus can be applied to different scenerios. I
knew that math was used to figure out things like the pipeline project, but
until now I would not have considered using a calculus approach. I am a
biology major so this project did not interest me as much as other
optimization problems that can be related to biological fields.
I have realized that calculus is much more involved than other math
classes I have taken in the past. It requires a solid understanding of the
concepts as well as a good understanding of previous classes such as college
algebra and trigonometry.