Questions: A boat sails on a bearing of 74° for 123 miles and then turns and sails 223 miles on a bearing of 195°. Find the distance of the boat from its starting point. The distance is miles. (Round to the nearest integer as needed.)

Solution Steps

The boat sails on a bearing of \( 74^\circ \) for \( 123 \) miles. We can find the coordinates after this leg using the cosine and sine functions:

\[ x_1 = 123 \cdot \cos(74^\circ) \approx 33.9034 \] \[ y_1 = 123 \cdot \sin(74^\circ) \approx 118.2352 \]

Thus, the coordinates after the first leg are approximately \( (33.9034, 118.2352) \).

Next, the boat turns and sails on a bearing of \( 195^\circ \) for \( 223 \) miles. We calculate the new coordinates:

\[ x_2 = 223 \cdot \cos(195^\circ) \approx -215.4015 \] \[ y_2 = 223 \cdot \sin(195^\circ) \approx -57.7166 \]

The coordinates after the second leg are approximately \( (-215.4015, -57.7166) \).

To find the final coordinates of the boat, we sum the coordinates from both legs:

\[ \text{Final } x = x_1 + x_2 \approx 33.9034 - 215.4015 \approx -181.4981 \] \[ \text{Final } y = y_1 + y_2 \approx 118.2352 - 57.7166 \approx 60.5185 \]

Thus, the final coordinates are approximately \( (-181.4981, 60.5185) \).

The distance from the starting point can be calculated using the Pythagorean theorem:

\[ d = \sqrt{(-181.4981)^2 + (60.5185)^2} \approx 191 \]

Final Answer