Mountain Distance 859369

1. **State the problem:** We need to find the distance between Silver Mountain and Mount Clether given that Silver Mountain is 165 km west of Mount Clether at a bearing of 291°. 2. **Understanding bearings:** A bearing of 291° means the direction is measured clockwise from north. 291° is equivalent to 360° - 291° = 69° west of north. 3. **Set up the problem:** We can model the positions using vectors or trigonometry. Since Silver Mountain is 165 km away at a bearing of 291°, we can find the east-west and north-south components of this displacement. 4. **Calculate components:** - The west-east component (x) = $165 \times \sin(291^\circ)$ - The north-south component (y) = $165 \times \cos(291^\circ)$ Note: Since 291° is in the northwest quadrant, sine and cosine values will be negative or positive accordingly. 5. **Calculate sine and cosine:** - $\sin(291^\circ) = \sin(360^\circ - 69^\circ) = -\sin(69^\circ) \approx -0.9336$ - $\cos(291^\circ) = \cos(360^\circ - 69^\circ) = \cos(69^\circ) \approx 0.3584$ 6. **Calculate components:** - $x = 165 \times (-0.9336) = -154.14$ km (west) - $y = 165 \times 0.3584 = 59.14$ km (north) 7. **Distance between mountains:** Since Silver Mountain is west of Mount Clether, the direct distance is the magnitude of the vector difference. 8. **Calculate distance:** $$\text{distance} = \sqrt{x^2 + y^2} = \sqrt{(-154.14)^2 + (59.14)^2}$$ $$= \sqrt{23758.6 + 3497.5} = \sqrt{27256.1} \approx 165.1 \text{ km}$$ 9. **Round to nearest km:** The distance is approximately $165$ km. **Final answer:** The two mountains are about $165$ km apart.