I only need help with part c. I don't know if the cost function is correct or not. But for the total cost after all the math, I got $ 10,737,980.74. Please check if it is correct or not.

Pipeline Proiect The US. 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 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 7 miles directly west and then 18 miles south and finally 35 miles east to the refinery (see figure below). Cost for materials, labor and fees to run the pipeline across BLM ground is $250,000 per mile. Cost of drilling through the existing mountain would be a one-time cost of $3,200,000 on top of the normal costs ($250,000 per mile) 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 $400,000 and will delay the project by 3 months costing the company another $120,000 per month. For any pipeline run across private ground, your company incurs an additional $75,000 per mile cost for right-of-way 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: i) One running west, south and then east to the refinery. ii) One heading east through the mountain and then south to the refinery. b) Determine the cost of running the pipeline: i) The shortest distance across the private ground to the refinery. ii) Straight south across the private ground, then straight east to the refinery. c) Determine the cost function for the 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 find the optimal way to run the pipeline to minimize cost. Determine the length of pipe that runs across private land and how far from the refinery it starts running on BLM land. Determine the angle at which your optimal path leaves the well. 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. d) Include a computer generated graph of the optimal cost function, \"C(x)", for this pipeline for any configuration involving crossing some private ground as well as some BLM ground. Make sure to use the correct domain of the function to scale your axes appropriately and to label the minimum point. BLM Ground 18 miles Private Ground BLM Ground 35 miles, BLM Ground Write up a report of your nding to submit to your company's CEO. This report should include all steps for any math used to make determinations asked for above as well as statements as to the costs to be incurred by each scenario. Include any appropriate figures to make each scenario clear. Label the scenarios as they are labeled above. Expectations: . All of the mathematical steps will be shown and will be correct - Use exact values, rounding costs to the nearest cent only after the final calculation - Each pipeline scenario will be clearly labeled on a computer generated diagram showing the path that is being considered a Any variables you use will be clearly defined and shown In your diagrams - The cost function graph will be accurately generated using mathematical software - A typed explanatory narrative with correct spelling and grammar will allow the reader to follow your process. Hand-written work will not be graded. The only portion of the report that does not need to be typed are the calculations. Your report should be an exposition of your finest work and presentation. Dear CEO, Cost = 325,000 (33.29) Take derivative and equate to zero to find the extrema values. Shortcut is in part d, where Explanation of part c: If you decide to use the optimal path for the pipeline construction, it will cost around I am writing to inform you of the completion of the BLM pipeline project. Using the Fixed C= $ 10,819,250 you graph the function and determine the highest point. $ 10,737,980.74, which will generate a savings of around $81,270 vs. the lowest costing Costs in the table below, I have determined the lowest path cost for the construction of the Case 2: Direct South-East path 0 = 5.85 . sec(@)tan(0) - 4.5 . sec.(0) project in any of the previous scenarios. pipeline. Below you will find the result of each scenario proposed Cost = (325,000)(18) + (250,000)(28) 0 = 5.85 . tan(0) - 4.5 . sec(0) Drill Savings over previous scenarios low cost - $81,270 C= 5,850,000 + 7,000,00 5.85 . tan(0) = 4.5 . sec(0) Angle $ 10,819,250 - $ 10,737,980.74 = $ 81,270 Fixed Costs: C= $ 12,850,000 sin(8) Drilling through mountain $3,200,000 cos (B) sec (8) 5.85 Research for drilling through the mountain $400,000 sin (8) cos ( 8 ) _ cos(@) = 5.85 18 miles Distance A Statement: Company costs for 3 months delay due to research $360,000 Of the above routes, the route which is the most affordable would be running in sin (0) = 0.7692 Pipeline cost per mile BLM grounds (x) $250,000 Private grounds in the shortest path, which would cost $ 10,819,250. 0 = sin-(0.7692) Distance B Refinery 0 = 0.8776 radians Pipeline cost per mile Private grounds (x) 325,000 10 Distance C 20 or . Cost for running in BLM grounds 0 = 50.28 Case 1: (West-South-East Path) Note: no mountain is traversed c. Cutting through Private grounds by some angle and proceeding to BLM grounds. Let: Cost = 250,000(Xw + Xs + Xe) Distance A = Da Explanation of part d: After many calculations, the optimal distance that runs through private land is: C= 250,000(7 + 18 + 35) Distance B = Db C = 250,000(60) = $ 15,000,000 Da = Cos(50.28) =0.639 28.167 miles y = (5.85 sec x +7 -4.5 tanx ) 0 Distance C = Dc (0.878,10.739) Case 2: (East-South Path) Note: mountain is traversed From the graph: Label Cost = 3,200,000 + 400,000 + 360,000 + 250,000(Xe + X's) After many other calculations, the optimal distance of running through BLM: C= 3,960,000 + 250,000(28 + 18) Da = cos (0) Db = 28 - 18 . tan(50.28) = 14.625 miles C= 3,960,000 + 11,500,000 De = 18 . tan(0) C = $ 15,460,000 Db= 28 - Dc = 28 - 18 . tan(0) Cost function would then be: b. Cost for running in Private grounds: = (325,000) Da + (250,000)Db (0.878, 10.738) Case 1: Shortest Path s(en-+ 7,000,000 - 4,500,000 . tan(@) D's = 28:+ 172 You can separate the zeros since they are very bulky and take up space. Ds = \\283+ V18 C= 1,000,000 (7 + 5.85 . sec(0) - 4.5 . tan()) Ds = 33.29 miles