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}} =