MATH 1210SIGNATURE PROJECT

By Sarah Anderson

Gas Pipeline Solution

The U.S. Interior Secretary recently approved drilling of natural gas wells near Vernal, Utah. Your company has begun drilling and established a high-producing 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 right-of-way 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 1 mile directly west and then 5 miles south and finally 21 miles east to the refinery. Cost for materials, labor and fees to run the pipeline across BLM ground is $300,000 per mile. For any pipeline run across private ground, your company incurs an additional $200,000 per mile cost for right-of-way fees. Cost of drilling through the existing mountain would be $500,000 on top of the normal costs of the material, labor and fees for 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 $100,000 and will delay the project by 3 months costing the company another $50,000 per month. Your company has asked you to do the following:

What We KnowCost:BLM Ground: $300,000 per milePrivate property: $300,000 per mile + $200,000 per mile=$500,000 per mileThrough the Mountain: $300,000 per mile+$500,000+$100,000+$150,000Distance:West around private property: 27 milesEast through mountain and around private property: 25 mileDiagonal across private land: 20.61552813 miles

𝒂𝟐+𝒃𝟐=𝒄𝟐

c5

20

𝟓𝟐+𝟐𝟎𝟐=𝒄𝟐

𝒄𝟐=𝟒𝟐𝟓𝒄=√𝟒𝟐𝟓

a) Determine the cost of running the pipeline strictly on BLM ground with two different scenarios:

1. heading east through the mountain and then south to the refinery

𝟑𝟎𝟎 ,𝟎𝟎𝟎 (𝟐𝟓)+𝟓𝟎𝟎 ,𝟎𝟎𝟎+𝟏𝟎𝟎 ,𝟎𝟎𝟎+𝟏𝟓𝟎 ,𝟎𝟎𝟎=$𝟖 ,𝟐𝟓𝟎 ,𝟎𝟎𝟎 .𝟎𝟎

a) Determine the cost of running the pipeline strictly on BLM ground with two different scenarios:

2. Running west, south and then east to the refinery.

𝟑𝟎𝟎 ,𝟎𝟎𝟎 (𝟐𝟕)=$𝟖 ,𝟏𝟎𝟎 ,𝟎𝟎𝟎 .𝟎𝟎

𝒂𝟐+𝒃𝟐=𝒄𝟐

c5

20

𝟓𝟐+𝟐𝟎𝟐=𝒄𝟐

𝒄𝟐=𝟒𝟐𝟓𝒄=√𝟒𝟐𝟓

b) Determine the cost of running the pipeline the shortest distance (straight line joining well to refinery across the private ground).

𝟑𝟎𝟎 ,𝟎𝟎𝟎 (𝟐𝟎 .𝟔𝟏𝟓𝟓𝟐𝟖𝟏𝟑 )+𝟐𝟎𝟎 ,𝟎𝟎𝟎 (𝟐𝟎 .𝟔𝟏𝟓𝟓𝟐𝟖𝟏𝟑 )=$𝟏𝟎 ,𝟑𝟎𝟕 ,𝟕𝟔𝟒 .𝟎𝟔

𝒄=𝟐𝟎 .𝟔𝟏𝟓𝟓𝟐𝟖𝟏𝟑

c) Determine the cost function for this pipeline for the configuration involving running from the well across the private ground at some angle and

intersecting the BLM ground to the south and then running east to the refinery. Use this function to determine the optimal place to run the pipeline to minimize cost. Clearly show all work including sketching the placement of the optimal pipeline. Make it very clear how you use your knowledge of calculus to determine the optimal placement of the pipeline. Draw a graph of this cost function and label the point of minimum cost.

20 miles-BLM Ground

5 m

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Pri

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P

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RPx 20-x

20 miles-BLM Ground

5 m

iles-

BLM

G

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RPx 20-x

𝑪 (𝒙 )=𝟑𝟎𝟎 ,𝟎𝟎𝟎 (𝟐𝟎−𝒙 )+𝟓𝟎𝟎 ,𝟎𝟎𝟎 (√𝒙𝟐+𝟐𝟓)𝑪 (𝒙 )=𝟔 ,𝟎𝟎𝟎 ,𝟎𝟎𝟎−𝟑𝟎𝟎 ,𝟎𝟎𝟎𝒙+𝟓𝟎𝟎 ,𝟎𝟎𝟎 (√𝒙𝟐+𝟐𝟓)

𝑪 (𝒙 )=𝟔 ,𝟎𝟎𝟎 ,𝟎𝟎𝟎−𝟑𝟎𝟎 ,𝟎𝟎𝟎𝒙+𝟓𝟎𝟎 ,𝟎𝟎𝟎 (𝒙𝟐+𝟐𝟓 )𝟏 /𝟐

(2x)

𝑪′ (𝒙 )=−𝟑𝟎𝟎 ,𝟎𝟎𝟎+𝟓𝟎𝟎 ,𝟎𝟎𝟎𝒙

√𝒙𝟐+𝟐𝟓

𝑿𝟐+𝒚 𝟐=𝒛𝟐

𝑿𝟐+𝟐𝟓=𝒛𝟐

𝒛=√𝒙𝟐+𝟐𝟓𝒛=√𝒙𝟐+𝟐𝟓z

𝑪′ (𝒙 )=−𝟑𝟎𝟎 ,𝟎𝟎𝟎+𝟓𝟎𝟎 ,𝟎𝟎𝟎𝒙

√𝒙𝟐+𝟐𝟓

−𝟑+𝟓 𝒙

√𝒙𝟐+𝟐𝟓=𝟎

𝟓 𝒙

√𝒙𝟐+𝟐𝟓=𝟑

𝟓 𝒙=𝟑(√𝒙𝟐+𝟐𝟓)𝟐𝟓𝒙𝟐=𝟗(𝒙𝟐+𝟐𝟓)

𝟑𝟎𝟎 ,𝟎𝟎𝟎=𝟑𝟓𝟎𝟎 ,𝟎𝟎𝟎=𝟓

𝟐𝟓𝒙𝟐=𝟗 𝒙𝟐+𝟐𝟐𝟓−𝟗 𝒙𝟐−𝟗 𝒙𝟐

𝟏𝟔𝒙𝟐=𝟐𝟐𝟓𝒙𝟐=

𝟐𝟐𝟓𝟏𝟔

√𝒙𝟐=√𝟐𝟐𝟓√𝟏𝟔

𝒙=𝟏𝟓𝟒

=𝟑 .𝟕𝟓

𝑪 (𝟎 )=(𝟐𝟎−𝟎 )𝟑+(√𝟎𝟐+𝟐𝟓)𝟓

Critical number at 3.75Endpoints at 0,20

𝟔𝟎+𝟐𝟓=𝟖𝟓=$𝟖 ,𝟓𝟎𝟎 ,𝟎𝟎𝟎 .𝟎𝟎

𝑪 (𝟐𝟎)=(𝟐𝟎−𝟐𝟎 )𝟑+ (√𝟐𝟎𝟐+𝟐𝟓 )𝟓𝟎+ (√𝟒𝟐𝟓 )𝟓=𝟏𝟎𝟑 .𝟎𝟕𝟕𝟔𝟒𝟎𝟔=$𝟏𝟎 ,𝟑𝟎𝟕 ,𝟕𝟔𝟒 .𝟎𝟔

5

𝟒𝟖 .𝟕𝟓+𝟑𝟏 .𝟐𝟓=𝟖𝟎=$𝟖 ,𝟎𝟎𝟎 ,𝟎𝟎𝟎 .𝟎𝟎

𝑪 (𝒙 )=(𝟐𝟎− 𝒙 )𝟑+(√𝒙𝟐+𝟐𝟓)𝟓

4 .75𝑚𝑖𝑙𝑒𝑠16.25𝑚𝑖𝑙𝑒𝑠

6.25𝑚𝑖𝑙𝑒𝑠

6.25 (500,000 )+16.25 (300,000 )=$ 8,000,000.00

25+14.0625=𝑧 2

𝑧=√39.0625𝑧=6.25

ReflectionI have to admit, I was pretty nervous about

taking calculus, especially online. I am surprised at how much I have learned this semester. It has been especially helpful to see the proofs of certain formulas so I don’t have to just trust that mathematicians know what they are doing. I have also found that I enjoy finding derivatives, now that I know all the rules to make them easier. Since my goal is to be high school math teacher, calculus could play a big part in that. I want to graduate with a level four teaching certificate which would qualify me to teach the higher level math classes. I know that math can be hard and boring for most high school students and so seeing calculus in the real world situations can help bring some enjoyment to class. I hope to be able to be a teacher who can bring that level of learning to the classroom.

Besides the fact that knowing calculus will help me to teach the higher level math classes, it could also come in handy in other areas. Knowing how to minimize cost could come in handy when doing home projects or any landscaping. I also like that I can find functions of things on my own and don’t have to rely on using a pre-determined function. It has been a stressful semester but the good kind of stress. I am looking forward to moving on in math and seeing where it can lead.