1. **State the problem:** We have a lighthouse at point L, a boat at point A which is 589 feet from L, and another point B on the shoreline. The angles of elevation to the lighthouse beacon from A and B are 6° and 3° respectively. We need to find the distance from A to B. 2. **Set up variables and diagram:** Let the height of the lighthouse be $h$. The horizontal distances are $AL = 589$ feet and $BL = x$ feet. We want to find $AB = |589 - x|$. 3. **Use the tangent of the angles:** The tangent of the angle of elevation equals the opposite side (height $h$) over the adjacent side (distance along shore). From point A: $$\tan(6^\circ) = \frac{h}{589} \implies h = 589 \tan(6^\circ)$$ From point B: $$\tan(3^\circ) = \frac{h}{x} \implies h = x \tan(3^\circ)$$ 4. **Set the two expressions for $h$ equal:** $$589 \tan(6^\circ) = x \tan(3^\circ)$$ 5. **Solve for $x$:** $$x = \frac{589 \tan(6^\circ)}{\tan(3^\circ)}$$ 6. **Calculate the tangent values:** $$\tan(6^\circ) \approx 0.1051, \quad \tan(3^\circ) \approx 0.05241$$ 7. **Substitute and compute $x$:** $$x = \frac{589 \times 0.1051}{0.05241} \approx \frac{61.85}{0.05241} \approx 1180.1$$ 8. **Find the distance $AB$:** $$AB = |589 - 1180.1| = 591.1$$ 9. **Round to nearest foot:** $$AB \approx 591 \text{ feet}$$ **Final answer:** The distance from point A to point B is approximately 591 feet.