1. The problem asks for the bearing of Kabale from Jinja and the shortest distance on the actual map. 2. Bearing is the direction from one point to another, measured clockwise from the north. 3. To find the bearing, we need the coordinates (latitude and longitude) of Jinja and Kabale. 4. The shortest distance on the Earth's surface between two points is the great-circle distance, calculated using the haversine formula: $$d = 2r \arcsin\left(\sqrt{\sin^2\left(\frac{\Delta\phi}{2}\right) + \cos(\phi_1)\cos(\phi_2)\sin^2\left(\frac{\Delta\lambda}{2}\right)}\right)$$ where $r$ is Earth's radius (approx. 6371 km), $\phi$ is latitude in radians, $\lambda$ is longitude in radians, and $\Delta$ denotes difference. 5. The initial bearing $\theta$ from point 1 to point 2 is given by: $$\theta = \arctan2\left(\sin(\Delta\lambda)\cos(\phi_2), \cos(\phi_1)\sin(\phi_2) - \sin(\phi_1)\cos(\phi_2)\cos(\Delta\lambda)\right)$$ 6. Convert $\theta$ from radians to degrees and normalize to $0^\circ$ to $360^\circ$. 7. Using approximate coordinates: - Jinja: $0.44^\circ$N, $33.20^\circ$E - Kabale: $1.25^\circ$S, $29.99^\circ$E 8. Convert degrees to radians: $\phi_1 = 0.44 \times \frac{\pi}{180} = 0.00768$ rad $\lambda_1 = 33.20 \times \frac{\pi}{180} = 0.5797$ rad $\phi_2 = -1.25 \times \frac{\pi}{180} = -0.0218$ rad $\lambda_2 = 29.99 \times \frac{\pi}{180} = 0.5236$ rad 9. Calculate differences: $\Delta\phi = \phi_2 - \phi_1 = -0.0295$ rad $\Delta\lambda = \lambda_2 - \lambda_1 = -0.0561$ rad 10. Calculate $a$ for distance: $$a = \sin^2\left(\frac{-0.0295}{2}\right) + \cos(0.00768)\cos(-0.0218)\sin^2\left(\frac{-0.0561}{2}\right) = 0.00037$$ 11. Calculate distance $d$: $$d = 2 \times 6371 \times \arcsin(\sqrt{0.00037}) = 2 \times 6371 \times 0.0192 = 244.7 \text{ km}$$ 12. Calculate initial bearing $\theta$: $$\theta = \arctan2(\sin(-0.0561)\cos(-0.0218), \cos(0.00768)\sin(-0.0218) - \sin(0.00768)\cos(-0.0218)\cos(-0.0561)) = \arctan2(-0.0561, -0.0211) = -1.95 \text{ rad}$$ 13. Convert $\theta$ to degrees and normalize: $$\theta_{deg} = (-1.95 \times \frac{180}{\pi}) + 360 = 271.8^\circ$$ 14. Interpretation: The bearing from Jinja to Kabale is approximately $271.8^\circ$, meaning almost due west. Final answers: - Shortest distance: approximately 245 km - Bearing from Jinja to Kabale: approximately $272^\circ$ (west)