1. **Problem 1: Total prize money distribution** The winner receives half of the total prize money, and the second runner-up receives one-fourth of the winner's amount. The winner's prize is given as 6000. 2. **Formula and explanation:** Let the total prize money be $T$. Winner's share = $\frac{1}{2}T = 6000$. Second runner-up's share = $\frac{1}{4} \times 6000 = 1500$. 3. **Calculate total prize money:** Since the winner gets half, $T = 2 \times 6000 = 12000$. 4. **Answer for Problem 1:** The total prize money distributed is $12000$. --- 5. **Problem 2: Distance from your ear to the top of the tree** You are 120 ft away from the base of a 50 ft tall tree. We want the straight-line distance from your ear to the top of the tree. 6. **Formula and explanation:** This forms a right triangle with legs 120 ft (horizontal distance) and 50 ft (vertical height). Use the Pythagorean theorem: $$d = \sqrt{120^2 + 50^2}$$ 7. **Calculate distance:** $$d = \sqrt{14400 + 2500} = \sqrt{16900} = 130$$ 8. **Answer for Problem 2:** The distance is approximately 130 feet. --- 9. **Problem 3: Total money spent on Routes 3 and 4** Route 3 costs 17 euros per person, Route 4 costs 35 euros per person. 10. **Calculate total cost:** $$17 + 35 = 52$$ 11. **Answer for Problem 3:** Total money spent is 52 euros.