Let me summarize the given situation and define the variables:

• The river is $\frac{3}{4}$ miles wide.
• The factory is 8 miles downstream.
• The cost per foot for laying power lines over land is $24/foot.
• The cost per foot for laying power lines under water is $30/foot.
• The total cost for the project is $1,103,149.53.

Step 1: Understand the Problem

From the problem, there are two different paths where the power lines can be laid:

1. Underwater across the width of the river for a distance of $x$ (in miles).
2. On land for a remaining distance, covering the rest of the 8 miles minus $x$ along the riverbank.

Step 2: Set Up the Cost Function

Convert all distances to feet:

• 1 mile = 5280 feet, so:
  • The width of the river: $\frac{3}{4} \times 5280 = 3960$ feet.
  • The total downstream distance (8 miles): $8 \times 5280 = 42,240$ feet.

Let $x$ be the horizontal distance underwater (in miles). We need to compute the total cost of laying power lines.

• Underwater segment cost: The underwater path has a length of $\sqrt{x^2 + (0.75)^2}$ miles (since it's the hypotenuse of a right triangle with the river width and $x$). In feet, this is: $\text{underwater distance (feet)} = 5280 \times \sqrt{x^2 + (0.75)^2}$ The cost for laying this underwater segment is: $30 \times 5280 \times \sqrt{x^2 + 0.75^2}$

• Land segment cost: The remaining land distance is $8 - x$ miles, or $5280 \times (8 - x)$ feet. The cost for laying the power lines on land is: $24 \times 5280 \times (8 - x)$

Step 3: Total Cost

The total cost of the project is given by: $30 \times 5280 \times \sqrt{x^2 + 0.75^2} + 24 \times 5280 \times (8 - x) = 1,103,149.53$ Simplify: $158400 \times \sqrt{x^2 + 0.75^2} + 126720 \times (8 - x) = 1,103,149.53$

Step 4: Solve for $x$

We can solve this equation numerically for $x$. Let me calculate the value of $x$.The value of $x$, the horizontal distance underwater, is approximately $2.200$ miles.

Would you like further details on how this was calculated or any other clarifications?

Here are 5 follow-up questions to consider:

1. How would the total cost change if the underwater cost per foot was reduced to $25?
2. What is the land distance that the power lines will cover based on this solution?
3. How would the solution change if the river were wider, say 1 mile instead of $\frac{3}{4}$ miles?
4. Can you derive a general formula to determine the cost based on different river widths and downstream distances?
5. What are the practical challenges in laying power lines underwater versus on land?

Tip: When solving real-world problems, setting up equations based on distances and costs is key to finding the optimal solution.