1. **State the problem:** A boat starts at point A, 994 feet from the lighthouse base L. The angle of elevation to the lighthouse beacon from A is 12°. Later, at point B (between A and L), the angle of elevation is 3°. We need to find the distance from A to B. 2. **Set up the problem:** Let the height of the lighthouse be $h$ feet. Let the distance from B to L be $x$ feet. Since A is 994 feet from L, the distance from A to B is $994 - x$. 3. **Use the tangent function for right triangles:** From point A: $$\tan(12^\circ) = \frac{h}{994}$$ From point B: $$\tan(3^\circ) = \frac{h}{x}$$ 4. **Express $h$ from both equations:** From A: $$h = 994 \tan(12^\circ)$$ From B: $$h = x \tan(3^\circ)$$ 5. **Set the two expressions for $h$ equal:** $$994 \tan(12^\circ) = x \tan(3^\circ)$$ 6. **Solve for $x$:** $$x = \frac{994 \tan(12^\circ)}{\tan(3^\circ)}$$ 7. **Calculate the tangent values:** $$\tan(12^\circ) \approx 0.2126$$ $$\tan(3^\circ) \approx 0.05241$$ 8. **Calculate $x$:** $$x = \frac{994 \times 0.2126}{0.05241} \approx \frac{211.3}{0.05241} \approx 4033.3$$ 9. **Find the distance from A to B:** $$994 - 4033.3 = \cancel{994} - 4033.3 = -3039.3$$ Since $x$ is greater than 994, point B is actually farther from L than A, so the distance from A to B is: $$4033.3 - 994 = 3039.3$$ 10. **Final answer:** The distance from point A to point B is approximately **3039.3 feet**.