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

Route 3: I think the hypotenuse is wrong in the text

d = d1-d3

cost = 2.7 x + 4.7 [(d-x)^2 +d2^2 ]^.5

dc/dx = 2.7 +4.7(1/2)[(d-x)^2 +d2^2 ]^-.5 *2(d-x)(-1)

now set that to zero and solve for x

Report:

Subject: Designing a Pipeline with Minimum Cost

Date: [Date]

To: [Supervisor]

From: [Your Name]

Introduction:

The objective of this investigation is to determine the most cost-effective route to connect an existing oil well located at Point A with a new well at Point B in an oil fertile area, considering various constraints and costs associated with pipeline installation. The attached map, a section of a U.S. Geological Survey contour map, provides an overview of the area, including a wetland region outlined. Our analysis aims to identify the route that incurs the least cost by considering multiple pipeline routes and their associated costs.

Assumptions:

To simplify our analysis, we will assume that the wetland separating A and B is rectangular in shape. We need to determine the dimensions of this rectangle to aid in our calculations. From the map, we can measure the farthest points north, south, east, and west of the wetland region to define the boundaries.

Considering Possible Routes:

To reduce the number of paths to consider, we need to evaluate the feasibility of various routes and identify potential constraints. Some factors to consider are as follows:

1. Path around the swamp:
- North of Point A: It is important to assess if a route with only one elbow joint provides a feasible option.
- South of Point B: We need to determine if this direction offers a viable path.

2. Possible Straight Routes:
- Due south of Point A: Analyze if it is possible to proceed only on normal terrain.
- Due north or south of Point B: Evaluate the costs associated with these routes.

Analyzing Costs:

To determine the minimum-cost route, we need to calculate the costs associated with different paths. The costs include the pipe, installation, and any additional expenses for work in the wetland area.

1. Cost of Pipe:
- Two-inch coated pipe is required at a cost of $1.50 per foot.

2. Installation Costs:
- Installation on normal terrain costs $1.20 per foot.
- Installation in the wetland area incurs an additional cost of $60 per hour, using a special that can dig approximately 300 feet of trench in a 10-hour day.

3. Elbow Joint:
- A maximum of one elbow joint may be used, with no extra cost associated.

Evaluation and Selection:

Based on our analysis, we will calculate the costs for multiple routes and select the one with the least cost. One potential option to consider is a route that starts southeast of Point A on the wetland terrain and then turns due south on normal terrain towards Point B.

To mathematically justify our selection, we will employ techniques from differential and integral calculus. By applying optimization principles, we can find the critical points on cost functions and determine the minimum cost route.

Conclusion:

Through thorough analysis and calculations, we aim to identify the pipeline route with the minimum cost to connect the existing well (Point A) with the new well (Point B) in the oil fertile area. By considering the constraints, costs, and applying mathematical optimization techniques, we will present the most cost-effective route to our supervisor.

Please let me know if you require any additional information or clarification.

Best regards,

[Your Name]