To create the mathematical formulation for the given pipeline project, we need to follow the rubric from the assessment document you provided. Here's a step-by-step approach to the formulation:

3. Mathematical Formulation

To model the problem mathematically, we need to:

1. Identify important elements of the problem:

   • The pipeline is to be laid from point A (7 km offshore) to point C (10 km along the shore).
   • The cost per kilometer for onshore and underwater parts of the pipeline is given as $p per km and $q per km, respectively.
   • Point B onshore is such that line AB is perpendicular to the shore BC.

2. Understand the problem's mathematical concepts:

   • The goal is to minimize the cost of the pipeline construction.
   • The total distance traveled by the pipeline can be split into two parts: one onshore (from B to C) and one underwater (from A to B).

3. Use appropriate mathematical terminology and methods:

   • Let the distance from A to B (underwater part) be denoted by $x$ (km).
   • The length of the onshore part of the pipeline, from B to C, is then $10 - x$ km.
   • The total cost function for the pipeline construction can be expressed as:

$\text{Total Cost} = p \times (10 - x) + q \times x$

Where:

• $p$ is the cost per km onshore,
• $q$ is the cost per km underwater,
• $x$ is the distance of the underwater pipeline, and
• $10 - x$ is the distance of the onshore pipeline.

4. Objective:
   • The objective is to minimize the total cost function with respect to $x$, where $0 \leq x \leq 7$ (since $A$ is 7 km offshore and $B$ lies along the shore).

This formulation identifies the mathematical relationships between the pipeline's layout and the costs of construction, which sets up the problem for the next steps of solution.

Would you like to move on to solving this or have further details on any specific section? Here are a few questions for further exploration:

1. How can we differentiate the cost function to find its minimum?
2. What are the implications of different values of $p$ and $q$ on the cost?
3. How does the geometry of the problem affect the total cost?
4. Would the model change if the pipeline could be laid at an angle rather than perpendicular to the shore?
5. How would changing the distance from the shore to point A affect the overall cost?

Tip: Always verify if the units for distance and cost are consistent throughout the calculation to avoid errors in the final result!