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To solve this problem, we need to minimize the cost of laying the cable by determining the optimal point $ P $ on the coast where the cable should reach land.

### Variables and Setup
- $ x $: The distance from the point $ P $ to the point $ E $ along the coast.
- The distance from $ P $ to the power relay station $ S $ (diagonally across the water) can be determined by the Pythagorean theorem as $ \sqrt{x^2 + 3000^2} $, since the distance from $ E $ to $ S $ across the water is 3000 feet.
- The total length of the cable laid on land is $ 10,000 - x $, because the total distance from the land point directly opposite the island to point $ P $ is 10,000 feet.

### Costs
- Cost of laying the cable on land: $1.50 per foot.
- Cost of laying the cable under water: $2.20 per foot.

### Total Cost Function
The total cost $ C(x) $ consists of two parts:
1. The cost of laying the cable on land from $ P $ to the point directly opposite the island, which is $ 1.50 \times (10,000 - x) $.
2. The cost of laying the cable under water from $ S $ to $ P $, which is $ 2.20 \times \sqrt{x^2 + 3000^2} $.

Thus, the total cost function is:
\[
C(x) = 1.50 \times (10,000 - x) + 2.20 \times \sqrt{x^2 + 3000^2}
\]

### Minimization
To find the value of $ x $ that minimizes the total cost, we need to take the derivative of $ C(x) $, set it equal to zero, and solve for $ x $.

1. **Derivative of $ C(x) $:**
   \[
   C'(x) = -1.50 + 2.20 \times \frac{x}{\sqrt{x^2 + 3000^2}}
   \]
   
2. **Setting $ C'(x) = 0 $ to find the critical points:**
   \[
   -1.50 + 2.20 \times \frac{x}{\sqrt{x^2 + 3000^2}} = 0
   \]
   
   Simplifying:
   \[
   2.20 \times \frac{x}{\sqrt{x^2 + 3000^2}} = 1.50
   \]
   \[
   \frac{x}{\sqrt{x^2 + 3000^2}} = \frac{1.50}{2.20} = \frac{15}{22}
   \]
   
   Now, square both sides:
   \[
   \frac{x^2}{x^2 + 3000^2} = \left(\frac{15}{22}\right)^2 = \frac{225}{484}
   \]
   
   Multiply both sides by $ x^2 + 3000^2 $:
   \[
   x^2 = \frac{225}{484}(x^2 + 3000^2)
   \]
   
   Expand and solve for $ x $.

I'll calculate this next to determine the value of $ x $.The positive solution to the equation is approximately $ x = 2796 $ feet.

This is the distance from point $ E $ to point $ P $ that will minimize the cost of laying the cable.

Would you like further clarification or details on any steps?

Here are 5 related questions to expand on the concept:
1. How would the total cost change if the price per foot under water increased?
2. Can you calculate the total cost for $ x = 2796 $ feet?
3. What if the distance across the water from $ E $ to $ S $ was different—how would that impact the solution?
4. How does the length of cable on land affect the total cost?
5. What happens if the costs of laying cable on land and under water are equal?

**Tip:** In optimization problems, always ensure you check for the minimum or maximum by either analyzing the second derivative or confirming the context of the problem!

Minimizing the cost of laying cable by determining the optimal point P on the coast where the cable should reach land. The cost of running the cable on land is $1.50 per running foot, and the cost of running the cable under water is $2.20 per running foot.

Learn how to minimize the cost of laying a cable by solving an optimization problem. This involves using derivatives and the Pythagorean theorem. The total cost function is derived based on the cost per running foot on land and under water, suitable for Grades 11-12.

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