To solve this problem, let's analyze each component based on the information provided in the image.

Problem Breakdown

1. Variables and Distance Measurements:

   • Let $x$ represent the horizontal distance (in km) from the point directly across the river to point $P$.
   • The total distance from the refinery to the storage tanks along the riverbank is $5$ km.

2. Pipe Costs and Distances:

   • $m = 5 - x$ km is the distance over land from the refinery to point $P$ along the north bank of the river.
   • $n = \sqrt{x^2 + 1}$ km is the underwater distance from point $P$ to the storage tanks across the river.

3. Cost Function:

   • Laying pipe over land costs $300,000 per km.
   • Laying pipe underwater costs $600,000 per km.

4. Objective:

   • Minimize the total cost $C(x)$ of laying the pipe.

Step 1: Formulate the Cost Function

The cost function $C(x)$ in units of $100,000 is given by: $C(x) = 3(5 - x) + 6\sqrt{x^2 + 1}$

Breaking it down:

• $3(5 - x)$ accounts for the land cost (in units of $100,000).
• $6\sqrt{x^2 + 1}$ accounts for the underwater cost (in units of $100,000).

Filling in Step 2

The cost function $C(x)$ is: $C(x) = 15 - 3x + 6\sqrt{x^2 + 1}$

This completes Step 2 with the cost function set up for further analysis, such as finding the derivative to minimize $C(x)$.

Would you like details on how to differentiate this function to find the minimum cost, or have any other questions?

Further Exploration

1. How to find the derivative $C'(x)$ and solve for the critical points.
2. Analysis of the critical points to determine the minimum cost.
3. Calculating $C(x)$ for specific values of $x$ to understand cost variations.
4. Explanation of why certain values of $x$ may be optimal in terms of cost.
5. Discussion on the real-world implications of pipeline placement and cost optimization.

Tip: Remember, in optimization problems, always confirm that the solution provides a minimum by checking the second derivative or testing values around the critical point.