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