Math Problem Statement

A company, drilling 7 kilometres offshore for oil, wants to run a pipeline from the drill site, A, to an oil storage facility, C, onshore. It cost $p per kilometre to run the pipeline onshore, and $q per kilometre to run it underwater. B is a point onshore such that AB is perpendicular to the straight shoreline BC. As a junior engineer on the project, you are required to: (a) model mathematically the cost of running the pipeline and to use the model to determine how to run the pipeline to minimise cost. (b) Use the model to determine whether the contract to run the pipeline should be awarded to either Company A or Company B below. Company A (p = US$415,000 and q = US$635,000) Company B (p = US$390,000 and q = US$650,000) (c) State the Length of the underwater pipeline and the length of the onshore pipeline for minimum cost for the selected company.

Solution

It seems you've uploaded a set of documents related to a project that involves determining the cost of running a pipeline offshore and onshore. The first image appears to show a rubric for evaluating the project, while the second image presents a detailed project description, including the geometric and cost aspects of the pipeline construction.

Here’s a summary of what needs to be done for this project:

  1. Mathematical Model of the Pipeline Costs: You are asked to formulate the mathematical cost model to represent the running costs for both onshore and underwater pipelines. This would involve using the cost per kilometer for each type of pipeline (onshore and underwater) and the distances they need to cover.

  2. Determine the Optimal Pipeline Path: You need to find the distance for both the underwater and onshore segments that minimizes the total cost.

  3. Comparing Costs for Two Companies: After modeling the costs, you'll calculate the total cost for both companies (A and B) using their respective cost rates for both the onshore and underwater pipelines. Then, you must determine which company offers the best value based on their rates and the length of the pipelines.

  4. Specific Deliverables: You’ll be asked to find the length of the underwater pipeline and the onshore pipeline to minimize the cost for the selected company.

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Math Problem Analysis

Mathematical Concepts

Optimization
Geometry
Cost Analysis
Trigonometry

Formulas

Total Cost = p * length_onshore + q * length_underwater
Pythagorean Theorem: a^2 + b^2 = c^2

Theorems

Optimization Theorem
Pythagorean Theorem

Suitable Grade Level

Grades 11-12