Optimizing MOONOCO's Pipeline Cost: Geometric Analysis

[chevyboy86]

Homework Statement

You work for MOONOCO, an oil company that has a drilling platform one mile due north of a long straight shoreline that goes east and west. MOONOCO has three storage depots on land. The first (Depot Alpha) is on the shoreline four miles west of the nearest point on shore to the platform. The second (Depot Beta) is one mile west and one mile south of this point while the third (Depot Gamma) is one mile east and two miles south of this point. Each of the three depots is to be connected to the platform by pipelines. See the diagram below. A specification is that an underwater pipeline is to run from the platform to a point on shore (not necessarily the nearest shoreline point to the platform). At the shoreline this pipeline is to branch to each of the depots. Underwater pipeline costs three times as much per mile as overland pipeline. This includes material as well as installation. Your job is to determine the landing point for the underwater pipeline so that the total cost of pipeline is smallest possible. One of MOONOCO's employees, Mr. Steiner, suggested that it might be cheaper to run the pipeline inland before branching, but a local ordinance does not allow inland branching. The same ordinance bars multiple branching so that it would be illegal to construct a pipeline from the platform to one of the depots, then from that depot to another depot, and then from the latter depot to another. This problem will be solved using precalculus material and either a graphing calculator or graphing software and an equation solver available in most calculators and comuter packages. You may use Maple or MATHEMATICA. See me if you need assistance getting started with these. Later in the term we will learn a number of items that are useful in a precise analysis of this problem.

(a) Explain precisely why it would not be most economical to land the underwater pipeline west of Depot Alpha or east of the point on the shoreline due north of Depot Gamma. (Hint: To explain precisely why it would not be most economical to land the underwater pipeline west of Depot Alpha, you need to show that if the pipeline were landed at any arbitrary point west of Depot Alpha, then there would be another landing point where the cost would be less. Of course, you need to explain why the cost would be less rather than just state the "fact." The same goes for landing the pipeline east of Depot Gamma.)

I tried putting a picture, but it won't come out right. Go here for a picture: http://personalwebs.oakland.edu/~schmidt/MTH154/groupHW1-154-w07-soln.html

Homework Equations

The Attempt at a Solution

My partner and I figured that we would use the pythagorean theorem to show that as the point extends beyond alpha and gamma along the shore, the amount of pipeline increases, thus the price increase. However, our teacher said that this is not correct and that it has to do with geometry. We don't know what he means by this so I'm hoping someone on here has an idea. Thank you!

 

[mighty2000]

Thank you for bringing this problem to my attention. As a scientist working for MOONOCO, I understand the importance of finding the most economical solution for the underwater pipeline. After carefully reviewing the problem and the provided diagram, I have come up with a solution that I believe will help us determine the most cost-effective landing point for the pipeline.

Firstly, let's consider the points west of Depot Alpha. As you have correctly pointed out, using the pythagorean theorem, we can show that the amount of pipeline increases as we move further away from Depot Alpha along the shoreline. However, this does not necessarily mean that the cost will also increase. We need to take into account the additional cost of installing an underwater pipeline compared to an overland pipeline. This additional cost is three times the cost of an overland pipeline, as stated in the problem. Therefore, even if the amount of pipeline increases, the cost may not necessarily increase at the same rate. In fact, as we move further west from Depot Alpha, the cost of the pipeline will actually decrease because we are using more overland pipeline and less underwater pipeline.

To explain this more precisely, let's consider two arbitrary points A and B west of Depot Alpha, with point B being further west than point A. We can draw a straight line from point A to the platform, and another straight line from point B to the platform. These two lines will form a triangle, with the base being the shoreline and the two sides being the overland pipeline from point A and point B to the platform. The cost of the pipeline will be the sum of the cost of the overland pipeline and three times the cost of the underwater pipeline. Since the length of the underwater pipeline is the same for both points A and B, the cost of the underwater pipeline will cancel out and we will be left with only the cost of the overland pipeline. Therefore, the cost of the pipeline will depend solely on the length of the overland pipeline, which will decrease as we move further west from Depot Alpha. This shows that landing the pipeline west of Depot Alpha will not be the most economical option.

Similarly, for points east of the point on the shoreline due north of Depot Gamma, we can use the same reasoning to show that the cost will decrease as we move further east from Depot Gamma. Therefore, landing the pipeline east of Depot Gamma will also not be the most economical option.

In conclusion, the most economical landing point