1. **State the problem:** We have a cliff 100m high with a lighthouse 30m tall on top. A boat sees the top of the lighthouse at an angle of elevation of 20°. 2. **Goal:** Calculate the distance from the boat to the base of the cliff (horizontal distance). 3. **Formula used:** In a right triangle, the tangent of an angle equals the opposite side over the adjacent side: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ 4. **Apply the formula:** The total height (opposite side) is $100 + 30 = 130$ meters. 5. Let the distance from the boat to the cliff base be $d$. Then: $$\tan(20^\circ) = \frac{130}{d}$$ 6. Solve for $d$: $$d = \frac{130}{\tan(20^\circ)}$$ 7. Calculate $\tan(20^\circ)$ (approximate): $$\tan(20^\circ) \approx 0.3640$$ 8. Substitute: $$d = \frac{130}{0.3640} \approx 357.14$$ **Final answer:** The distance from the boat to the base of the cliff is approximately **357.14 meters**.