1. **Problem statement:** Ibraheem hikes from Gator Swamp to Champion Lookout. Gator Swamp is 4 km from Old Town Road along Route 67 at a bearing of 15°. The hiking trail from Gator Swamp to Champion Lookout is at a bearing of 45° and slopes upward at an angle of elevation of 20°. We need to find: a) The horizontal distance between Gator Swamp and Champion Lookout on a map. b) The actual distance Ibraheem walks. c) The elevation difference between Champion Lookout and Gator Swamp. d) The average angle of elevation of the section of Old Town Road shown. 2. **Key formulas and concepts:** - Bearings are angles measured clockwise from north. - Horizontal distance on a map ignores elevation. - Actual distance along the hiking trail includes elevation. - Elevation difference = actual distance \( \times \sin(\text{angle of elevation}) \). - Horizontal distance along the trail = actual distance \( \times \cos(\text{angle of elevation}) \). - To find horizontal distance between two points, use vector components and the law of cosines or coordinate geometry. 3. **Step a) Horizontal distance between Gator Swamp and Champion Lookout on a map:** - Route 67 from Old Town Road to Gator Swamp: 4 km at 15° bearing. - Hiking trail from Gator Swamp to Champion Lookout: bearing 45°. Let the horizontal distance from Gator Swamp to Champion Lookout be \(d\). The angle between Route 67 and the hiking trail on the map is \(45^\circ - 15^\circ = 30^\circ\). Using the law of cosines for the triangle formed by Old Town Road (O), Gator Swamp (G), and Champion Lookout (C): $$OG = 4 \text{ km},$$ $$\angle GOC = 30^\circ,$$ We want to find the horizontal distance \(GC = d\). Since the horizontal distance from Old Town Road to Champion Lookout is unknown, we consider the horizontal projection of the hiking trail. 4. **Step b) Actual distance Ibraheem walks:** - The hiking trail slopes upward at 20°. - Let the horizontal distance from Gator Swamp to Champion Lookout be \(d\). - Actual distance walked \(= \frac{d}{\cos 20^\circ}\). 5. **Step c) Elevation difference:** - Elevation difference \(= \text{actual distance} \times \sin 20^\circ = \frac{d}{\cos 20^\circ} \times \sin 20^\circ = d \tan 20^\circ\). 6. **Step d) Average angle of elevation of Old Town Road section:** - Old Town Road is flat, so angle of elevation is 0°. 7. **Calculations:** - Coordinates for Gator Swamp relative to Old Town Road: $$x_G = 4 \sin 15^\circ = 4 \times 0.2588 = 1.0352 \text{ km}$$ $$y_G = 4 \cos 15^\circ = 4 \times 0.9659 = 3.8636 \text{ km}$$ - Let horizontal distance from Gator Swamp to Champion Lookout be \(d\). - Coordinates for Champion Lookout relative to Gator Swamp: $$x_C = d \sin 45^\circ = d \times 0.7071$$ $$y_C = d \cos 45^\circ = d \times 0.7071$$ - Coordinates for Champion Lookout relative to Old Town Road: $$x_{C, total} = x_G + x_C = 1.0352 + 0.7071 d$$ $$y_{C, total} = y_G + y_C = 3.8636 + 0.7071 d$$ - The horizontal distance between Old Town Road and Champion Lookout is along the line connecting these points. - The horizontal distance between Old Town Road and Champion Lookout is unknown, but we can find \(d\) by considering the triangle formed. - Since the problem does not give the distance from Old Town Road to Champion Lookout, we assume the horizontal distance between Gator Swamp and Champion Lookout is \(d\). - To find \(d\), we use the fact that the horizontal distance between Old Town Road and Champion Lookout is the straight line connecting their coordinates. - The horizontal distance between Old Town Road and Champion Lookout is: $$D = \sqrt{(1.0352 + 0.7071 d)^2 + (3.8636 + 0.7071 d)^2}$$ - Since the problem does not provide \(D\), we cannot solve for \(d\) numerically without additional data. **Therefore, we interpret the problem as asking for the horizontal distance between Gator Swamp and Champion Lookout on the map, which is the horizontal projection of the hiking trail.** - The horizontal distance between Gator Swamp and Champion Lookout is \(d\). - The actual distance Ibraheem walks is \(\frac{d}{\cos 20^\circ}\). - The elevation difference is \(d \tan 20^\circ\). 8. **Summary answers:** - a) Horizontal distance between Gator Swamp and Champion Lookout on the map is \(d\) km. - b) Actual distance walked is \(\frac{d}{\cos 20^\circ}\) km. - c) Elevation difference is \(d \tan 20^\circ\) km. - d) Average angle of elevation of Old Town Road is 0° because it is flat. **Note:** Without the actual distance \(d\) or additional data, we express answers in terms of \(d\).