1. **Problem statement:** An observer in a boat sees a tower on top of a cliff. The angle of elevation to the tower's top is 60° and to the cliff's top is 30°. The tower's height is 60 meters. Find the height of the cliff. 2. **Diagram and variables:** Let the height of the cliff be $h$ meters. The tower is on top of the cliff, so total height to tower top is $h + 60$ meters. 3. **Using trigonometry:** The observer's line of sight forms right triangles with the cliff and tower. - For the cliff top (angle 30°): $$\tan 30^\circ = \frac{h}{d}$$ where $d$ is the horizontal distance from observer to cliff base. - For the tower top (angle 60°): $$\tan 60^\circ = \frac{h + 60}{d}$$ 4. **Recall values:** $$\tan 30^\circ = \frac{1}{\sqrt{3}}, \quad \tan 60^\circ = \sqrt{3}$$ 5. **Express $d$ from first equation:** $$d = \frac{h}{\tan 30^\circ} = h \sqrt{3}$$ 6. **Substitute $d$ into second equation:** $$\tan 60^\circ = \frac{h + 60}{d} = \frac{h + 60}{h \sqrt{3}}$$ 7. **Plug in $\tan 60^\circ = \sqrt{3}$:** $$\sqrt{3} = \frac{h + 60}{h \sqrt{3}}$$ 8. **Multiply both sides by $h \sqrt{3}$:** $$h \sqrt{3} \times \sqrt{3} = h + 60$$ 9. **Simplify left side:** $$h \times 3 = h + 60$$ 10. **Solve for $h$:** $$3h - h = 60$$ $$2h = 60$$ $$h = 30$$ **Final answer:** The height of the cliff is **30 meters**.