1. **Problem Statement:** Kurt wants to sail from a marina to an island 15 km due east. He sails first on a heading of N 70° E, then on a heading of 120°. We need to find the total distance traveled before reaching the island. 2. **Understanding the headings:** - N 70° E means 70° east of north. - Heading 120° means 120° clockwise from north. 3. **Set up the triangle:** - The island is 15 km due east from the marina. - The first leg is along N 70° E. - The second leg is along heading 120°. 4. **Find the angle between the two legs:** - The first leg direction is 70° from north. - The second leg direction is 120° from north. - The angle between the two legs is $$120^\circ - 70^\circ = 50^\circ$$. 5. **Use Law of Cosines:** - Let the first leg be length $x$, the second leg be length $y$, and the direct distance (15 km) be side $c$ opposite the 50° angle. - The triangle formed has sides $x$, $y$, and $15$ km with included angle 50° between $x$ and $y$. 6. **Express the east displacement:** - The east component of the first leg is $x \sin 70^\circ$. - The east component of the second leg is $y \cos 30^\circ$ because heading 120° is 30° south of east. - Total east displacement must be 15 km: $$x \sin 70^\circ + y \cos 30^\circ = 15$$ 7. **Express the north displacement:** - The north component of the first leg is $x \cos 70^\circ$. - The north component of the second leg is $-y \sin 30^\circ$ (south is negative north). - Since the island is due east, total north displacement is zero: $$x \cos 70^\circ - y \sin 30^\circ = 0$$ 8. **Solve the system:** From north displacement: $$x \cos 70^\circ = y \sin 30^\circ$$ $$y = \frac{x \cos 70^\circ}{\sin 30^\circ}$$ Substitute into east displacement: $$x \sin 70^\circ + \frac{x \cos 70^\circ}{\sin 30^\circ} \cos 30^\circ = 15$$ Simplify: $$x \left(\sin 70^\circ + \frac{\cos 70^\circ \cos 30^\circ}{\sin 30^\circ}\right) = 15$$ Calculate values: $$\sin 70^\circ \approx 0.9397$$ $$\cos 70^\circ \approx 0.3420$$ $$\cos 30^\circ \approx 0.8660$$ $$\sin 30^\circ = 0.5$$ So: $$x (0.9397 + \frac{0.3420 \times 0.8660}{0.5}) = 15$$ $$x (0.9397 + 0.5927) = 15$$ $$x (1.5324) = 15$$ $$x = \frac{15}{1.5324} \approx 9.79$$ Then: $$y = \frac{9.79 \times 0.3420}{0.5} = 6.69$$ 9. **Total distance traveled:** $$9.79 + 6.69 = 16.48$$ km 10. **Final answer:** Kurt travels approximately **16.5 km** before reaching the island.