A common problem encountered by the oil industry is to determine the most cost-effective route to connect various wells in an oil fertile area. The attached map is a copy of a section of a U.S. Geographical Survey contour map with a wetland area outlined. An existing oil well is located approximately at the point labeled A. A new well is to be dug at point B, requiring that the pipeline installation company be given plans for laying connecting pipe from A to B. In consultation with the installation company, the following information has been obtained:
• Straight, two-inch coated pipe must be used, at a cost of $ 1.50 per foot.
• A maximum of one elbow joint may be used. The elbow joint may be fabricated with any angle measure and will not be included in the cost.
• On normal terrain, installation cost is $ 1.20 per foot.
• Installation in the wetland area requires the use of a special , at an additional installation cost of $ 60 per hour over and above normal installation cost.
• In a 10-hour day, this special can dig approximately 300 feet of trench.
Write up your investigation as a report to your supervisor, who should be able to understand your report without reference to this sheet. Demonstrate that you and your partner(s) have considered the costs of what you believe to be several viable pipeline routes connecting the existing well at A to the new well at B. Discuss your selection for the route that incurs the least cost and mathematically justify that your selection will give the least cost of all possible routes from A to B. Be sure to explain why minimum distance may not yield minimum cost.
Hints:
• Since the wetland separating A and B is very irregular in shape, simplify your work from the beginning by making some (slightly exaggerated) assumptions. You are to assume that the entire wetland is in the shape of a rectangle. How far to the east will your rectangle go? How far to the west? To the north? To the south?
• Reduce the number of paths you consider before calculating anything. Should you consider a path around the swamp to the north of A (with only one elbow joint)? What about to the south of B? Can you really go due south of A only on normal terrain?
• Do some measuring and calculating of costs of various paths, but be sure you consider the case where the pipeline is laid some distance southeast of A on wetland terrain and then turns due south on normal terrain to B. Perhaps this is where calculus comes in?
If you type in this :Engineering Applications in Differential and Integral Calculus* onto google on page four you will see the map. Please help
the title of this lab is designing a pipeline with minimum cost
Report: Designing a Pipeline with Minimum Cost
Problem Statement:
The objective of this investigation is to determine the most cost-effective route to connect an existing oil well located at point A to a new well to be dug at point B in an oil fertile area. The key factors for consideration in determining the route are the cost of the pipe, the number of elbow joints, and the terrain through which the pipe must be installed. Minimum distance may not necessarily yield minimum cost due to variations in installation costs and pipe expenses.
Assumptions:
To simplify the calculations, we will assume that the wetland area separating A and B can be approximated as a rectangle. We will determine the dimensions of this rectangle by considering the furthest extent to the east, west, north, and south.
Route Considerations:
Before calculating the costs of various paths, we will reduce the number of paths to consider. Based on the information provided, we can eliminate the option of a path around the swamp to the north of A since it would require an elbow joint. Similarly, going due south of A on normal terrain is not a feasible option due to a portion of wetland in that direction.
Considering these factors, we will focus on the possibility of laying the pipeline some distance southeast of A on wetland terrain and then turning due south on normal terrain to B.
Cost Analysis:
To calculate the cost of the pipeline, we need to consider the pipe expenses, installation costs, and any additional costs for the wetland area.
1. Pipe expenses:
The cost of using straight, two-inch coated pipe is $1.50 per foot. To calculate the length of the pipeline, we need to determine the distance between points A and B, taking into account the wetland area.
2. Installation costs:
On normal terrain, the installation cost is $1.20 per foot. However, in the wetland area, a special is required, which incurs an additional installation cost of $60 per hour, in addition to the normal installation cost.
3. Wetland area:
Since the wetland area is rectangular, we can determine the dimensions by measuring the furthest extent to the east, west, north, and south.
Mathematical Justification:
To mathematically justify that our selection will give the least cost among all possible routes from A to B, we will consider the option of laying the pipeline some distance southeast of A on wetland terrain and then turning due south on normal terrain to B.
The cost can be represented as a function of the position along the pipeline route. By using calculus, we can find the minimum cost by finding the minimum point of this function.
Next Steps:
To proceed with the calculations and determine the exact dimensions of the wetland area as well as the distance between points A and B, the provided map should be consulted. The map can be found by searching for "Engineering Applications in Differential and Integral Calculus" on Google, and accessing page four of the search results.
With this additional information and the mathematical justification, we can present a conclusive recommendation to our supervisor regarding the route that incurs the least cost for connecting the existing well at point A to the new well at point B.