Missing Distances C82Ea9

1. **State the problem:** We are given a traverse with six sides and their bearings. The lengths of sides AB and DE are missing. We need to calculate these missing distances. 2. **Understanding bearings and traverse:** Bearings are angles measured clockwise or counterclockwise from the north or south direction. To find missing distances, we use the fact that the traverse is a closed polygon, so the sum of the eastings (x-components) and northings (y-components) must be zero. 3. **Convert bearings to azimuths:** - AB: N 28° 15' E means azimuth $= 28.25^\circ$ - BC: S 80° 10' E means azimuth $= 180^\circ - 80.1667^\circ = 99.8333^\circ$ - CD: S 19° 47' W means azimuth $= 180^\circ + 19.7833^\circ = 199.7833^\circ$ - DE: S 31° 50' E means azimuth $= 180^\circ - 31.8333^\circ = 148.1667^\circ$ - EF: N 86° 42' W means azimuth $= 360^\circ - 86.7^\circ = 273.3^\circ$ - FA: N 16° 15' W means azimuth $= 360^\circ - 16.25^\circ = 343.75^\circ$ 4. **Set variables for missing distances:** Let $AB = x$ and $DE = y$. 5. **Calculate components for each side:** - $AB_x = x \times \sin(28.25^\circ)$ - $AB_y = x \times \cos(28.25^\circ)$ - $BC_x = 304.22 \times \sin(99.8333^\circ)$ - $BC_y = 304.22 \times \cos(99.8333^\circ)$ - $CD_x = 244.15 \times \sin(199.7833^\circ)$ - $CD_y = 244.15 \times \cos(199.7833^\circ)$ - $DE_x = y \times \sin(148.1667^\circ)$ - $DE_y = y \times \cos(148.1667^\circ)$ - $EF_x = 341.06 \times \sin(273.3^\circ)$ - $EF_y = 341.06 \times \cos(273.3^\circ)$ - $FA_x = 250.10 \times \sin(343.75^\circ)$ - $FA_y = 250.10 \times \cos(343.75^\circ)$ 6. **Calculate known components:** - $BC_x = 304.22 \times \sin(99.8333^\circ) \approx 304.22 \times 0.9848 = 299.66$ - $BC_y = 304.22 \times \cos(99.8333^\circ) \approx 304.22 \times (-0.1736) = -52.81$ - $CD_x = 244.15 \times \sin(199.7833^\circ) \approx 244.15 \times (-0.3420) = -83.52$ - $CD_y = 244.15 \times \cos(199.7833^\circ) \approx 244.15 \times (-0.9397) = -229.25$ - $EF_x = 341.06 \times \sin(273.3^\circ) \approx 341.06 \times (-0.9986) = -340.57$ - $EF_y = 341.06 \times \cos(273.3^\circ) \approx 341.06 \times (-0.0523) = -17.84$ - $FA_x = 250.10 \times \sin(343.75^\circ) \approx 250.10 \times (-0.2806) = -70.15$ - $FA_y = 250.10 \times \cos(343.75^\circ) \approx 250.10 \times 0.9598 = 239.95$ 7. **Write equations for closure:** - Sum of x-components $= 0$: $$x \sin(28.25^\circ) + 299.66 - 83.52 + y \sin(148.1667^\circ) - 340.57 - 70.15 = 0$$ - Sum of y-components $= 0$: $$x \cos(28.25^\circ) - 52.81 - 229.25 + y \cos(148.1667^\circ) - 17.84 + 239.95 = 0$$ 8. **Calculate sine and cosine for unknown sides:** - $\sin(28.25^\circ) \approx 0.4737$ - $\cos(28.25^\circ) \approx 0.8807$ - $\sin(148.1667^\circ) \approx 0.5299$ - $\cos(148.1667^\circ) \approx -0.8480$ 9. **Substitute values:** - X equation: $$0.4737x + 299.66 - 83.52 + 0.5299y - 340.57 - 70.15 = 0$$ Simplify constants: $$0.4737x + 0.5299y + (299.66 - 83.52 - 340.57 - 70.15) = 0$$ $$0.4737x + 0.5299y - 194.58 = 0$$ - Y equation: $$0.8807x - 52.81 - 229.25 - 0.8480y - 17.84 + 239.95 = 0$$ Simplify constants: $$0.8807x - 0.8480y + (-52.81 - 229.25 - 17.84 + 239.95) = 0$$ $$0.8807x - 0.8480y - 59.95 = 0$$ 10. **Solve the system:** From X equation: $$0.4737x + 0.5299y = 194.58$$ From Y equation: $$0.8807x - 0.8480y = 59.95$$ Multiply first equation by $0.8480$ and second by $0.5299$ to eliminate $y$: $$0.4737 \times 0.8480 x + 0.5299 \times 0.8480 y = 194.58 \times 0.8480$$ $$0.8807 \times 0.5299 x - 0.8480 \times 0.5299 y = 59.95 \times 0.5299$$ Calculate: $$0.4019x + 0.4497y = 165.11$$ $$0.4667x - 0.4497y = 31.75$$ Add equations: $$(0.4019 + 0.4667)x + (0.4497 - 0.4497)y = 165.11 + 31.75$$ $$0.8686x = 196.86$$ $$x = \frac{196.86}{0.8686} \approx 226.66$$ Substitute $x$ into first equation: $$0.4737(226.66) + 0.5299y = 194.58$$ $$107.39 + 0.5299y = 194.58$$ $$0.5299y = 87.19$$ $$y = \frac{87.19}{0.5299} \approx 164.53$$ 11. **Final answer:** - $AB = 226.66$ units - $DE = 164.53$ units