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To minimize the total construction costs of laying the pipe, we can break down this optimization problem with the following setup:

1. **Define Variables:**
   Let \( x \) be the distance down the shoreline from \( P \) where the pipe reaches land.

2. **Cost Factors:**
   - The cost of laying pipe in water is 1.5 times the cost of laying it on land.
   - The resort is 6 miles offshore from \( P \), and the fresh water source is 10 miles down the shoreline from \( P \).

3. **Distance Calculations:**
   - **Water Distance (to reach land):** The distance from the resort (offshore) to the point on land \( x \) miles down the shoreline is given by the hypotenuse of a right triangle with legs 6 miles (offshore distance) and \( x \) miles. So, the water distance is \( \sqrt{x^2 + 6^2} = \sqrt{x^2 + 36} \).
   - **Land Distance (to reach water source):** After reaching the shore \( x \) miles from \( P \), the remaining distance along the shore to the water source is \( (10 - x) \) miles.

4. **Cost Function:**
   Let \( C(x) \) be the total cost function for laying the pipe:
   \[
   C(x) = 1.5 \cdot \sqrt{x^2 + 36} + (10 - x)
   \]
   where \( 1.5 \cdot \sqrt{x^2 + 36} \) represents the cost of laying the pipe in water, and \( (10 - x) \) represents the cost of laying it on land.

5. **Optimization:**
   To find the optimal \( x \), we need to minimize \( C(x) \). This involves taking the derivative of \( C(x) \) with respect to \( x \), setting it equal to zero, and solving for \( x \).

   The derivative is:
   \[
   C'(x) = \frac{1.5 \cdot x}{\sqrt{x^2 + 36}} - 1
   \]
   Setting \( C'(x) = 0 \) for optimization:
   \[
   \frac{1.5 \cdot x}{\sqrt{x^2 + 36}} = 1
   \]
   Solving this equation will yield the optimal value of \( x \).

6. **Solution:**
   Solving the above equation gives:
   \[
   x = \frac{\sqrt{36}}{\sqrt{1.5^2 - 1}} = 

A small resort is situated on an island that lies exactly 6 miles from P, the nearest point to the island along a perfectly straight shoreline. 10 miles down the shoreline from P is the closest source of fresh water. If it costs 1.5 times as much money to lay pipe in the water as it does on land, how far down the shoreline from P should the pipe from the island reach land in order to minimize the total construction costs?

Learn how to minimize the construction cost for laying a pipe from an island to a water source 10 miles down the shoreline. This problem involves trigonometry and optimization techniques, including a cost function and derivative-based minimization, ideal for advanced calculus students.

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