1. **State the problem:** We need to find the distance between the boat and the top of the lighthouse. The lighthouse height is 31.7 m, and the horizontal distance from the boat to the base of the lighthouse is 0.5 km. 2. **Convert units:** Since the height is in meters and the horizontal distance is in kilometers, convert 0.5 km to meters: $$0.5 \text{ km} = 0.5 \times 1000 = 500 \text{ m}$$ 3. **Identify the triangle and use the Pythagorean theorem:** The boat, the base of the lighthouse, and the top of the lighthouse form a right triangle. The vertical side is 31.7 m, the horizontal side is 500 m, and the hypotenuse is the distance we want to find, call it $d$. The Pythagorean theorem states: $$d^2 = (\text{height})^2 + (\text{horizontal distance})^2$$ 4. **Substitute values:** $$d^2 = 31.7^2 + 500^2$$ 5. **Calculate squares:** $$31.7^2 = 1004.89$$ $$500^2 = 250000$$ 6. **Sum the squares:** $$d^2 = 1004.89 + 250000 = 251004.89$$ 7. **Find the distance $d$ by taking the square root:** $$d = \sqrt{251004.89}$$ 8. **Calculate the square root:** $$d \approx 501.00 \text{ m}$$ 9. **Round to the nearest meter:** $$d \approx 501 \text{ m}$$ **Final answer:** The distance between the boat and the top of the lighthouse is approximately 501 meters.