1. **State the problem:** We need to find the horizontal distance $d$ between the observation tower and the fire. 2. **Given data:** - Height of mountain = 500 m - Height of tower = 80 m - Total height = $500 + 80 = 580$ m - Angle of depression = $6^\circ$ 3. **Understanding the problem:** The angle of depression from the top of the tower to the fire is $6^\circ$. This angle is equal to the angle of elevation from the fire to the top of the tower (alternate interior angles). 4. **Set up the right triangle:** - Opposite side (vertical height) = 580 m - Adjacent side (horizontal distance) = $d$ - Angle = $6^\circ$ 5. **Use the tangent function:** $$\tan(6^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{580}{d}$$ 6. **Solve for $d$:** $$d = \frac{580}{\tan(6^\circ)}$$ 7. **Calculate $\tan(6^\circ)$:** Using a calculator, $\tan(6^\circ) \approx 0.1051$ 8. **Substitute and compute:** $$d = \frac{580}{0.1051} \approx 5522.36$$ 9. **Final answer:** The horizontal distance $d$ between the tower and the fire is approximately **5522.36 meters**.