A power line is to be constructed from a power station at point $A$ to an island at point $C$, which is $4 \mathrm{mi}$ directly out in the water from a point $B$ on the shore. Point $B$ is $8 \mathrm{mi}$ downshore from the power station at $A$. It costs $\$ 4200$ per mile to lay the power line under water and $\$ 3000$ per mile to lay the line under ground. At what point $S$ downshore from $A$ should the line come to the shore in order to minimize cost? Note that S could very well be B or A. (Hint: The length of $\operatorname{CS}$ is $\sqrt{16+x^{2}}$.) $S$ is $\square$ miles from $A$. (Round to two decimal places as needed.)

Step 1 ：The problem is asking to find the point S on the shore where the power line should come to minimize the cost. The cost of laying the power line under water is more expensive than laying it under ground. Therefore, we need to minimize the length of the power line under water.

Step 2 ：The total cost of the power line is the sum of the cost of the part from A to S and the cost of the part from S to C. The length of AS is x (the distance from A to S), and the length of SC is \(\sqrt{16 + (8-x)^2}\) (the distance from S to C, which is the hypotenuse of a right triangle with sides 4 and 8-x).

Step 3 ：The cost of AS is \(3000x\) (since it costs $3000 per mile), and the cost of SC is \(4200*\sqrt{16 + (8-x)^2}\) (since it costs $4200 per mile). Therefore, the total cost is \(3000x + 4200*\sqrt{16 + (8-x)^2}\).

Step 4 ：We need to find the value of x that minimizes this total cost. This is a calculus problem: we need to take the derivative of the total cost with respect to x, set it equal to zero, and solve for x. This will give us the value of x that minimizes the total cost.

Step 5 ：The critical point we found is \(8 - \frac{5\sqrt{6}}{3}\). This is the value of x that minimizes the total cost. However, we need to check if this value is within the valid range for x. Since x represents the distance from A to S, it must be between 0 and 8 (inclusive). If \(8 - \frac{5\sqrt{6}}{3}\) is not within this range, then the minimum cost occurs at either x = 0 or x = 8.

Step 6 ：Final Answer: The point $S$ is \(\boxed{8 - \frac{5\sqrt{6}}{3}}\) miles from $A$.