A boat sails on a bearing of $66^{\circ}$ for 138 miles and then turns and sails 213 miles on a bearing of $190^{\circ}$. Find th distance of the boat from its starting point.

The distance is $\square$ miles.
(Round to the nearest integer as needed.)

Step 1 ：Given that a boat sails on a bearing of \(66^{\circ}\) for 138 miles and then turns and sails 213 miles on a bearing of \(190^{\circ}\). We are to find the distance of the boat from its starting point.

Step 2 ：We can solve this problem using the law of cosines. The law of cosines states that for any triangle with sides of lengths a, b, and c and an angle γ between sides a and b, the following equation holds: \(c² = a² + b² - 2ab \cos(γ)\).

Step 3 ：In this case, the boat's journey forms a triangle with sides of lengths 138 miles and 213 miles, and an angle of 190° - 66° = 124° between them.

Step 4 ：We can use the law of cosines to find the length of the third side of the triangle, which represents the distance of the boat from its starting point.

Step 5 ：We need to convert the angle from degrees to radians before using it in the cosine function.

Step 6 ：Let's denote the sides of the triangle as follows: \(a = 138\), \(b = 213\), and the angle γ in degrees as \(γ_{\text{degrees}} = 124\).

Step 7 ：We convert the angle from degrees to radians: \(γ_{\text{radians}} = 2.1642082724729685\).

Step 8 ：We substitute these values into the law of cosines to find \(c^2\): \(c^2 = 97286.83240923825\).

Step 9 ：We take the square root of \(c^2\) to find the length of side c: \(c = 311.90837181652927\).

Step 10 ：We round the result to the nearest integer to get the final answer: \(c_{\text{rounded}} = 312\).

Step 11 ：Final Answer: The distance of the boat from its starting point is \(\boxed{312}\) miles.