A power station is on one side of a river that is 3/4 miles wide, and a factory is 8 miles downstream on the other side of the river. It costs $24 per foot to run power lines over land and $30 per foot to run them under water. The project's cost is $1,103,149.53. Find the length x labeled in the figure. (Enter your answers as a comma-separated list. Round your answers to three decimal places.)


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.

A power station is on one side of a river that is 3/4 miles wide, and a factory is 8 miles downstream on the other side of the river. It costs $24 per foot to run power lines over land and $30 per foot to run them under water. The project's cost is $1,103,149.53. Find the length x labeled in the figure. (Enter your answers as a comma-separated list. Round your answers to three decimal places.)

Solve a real-world optimization problem involving the cost of running power lines across a river. This detailed solution covers the Pythagorean theorem, distance conversions, and cost functions. Ideal for high school students learning geometry and optimization techniques.

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