Solution: The angle of elevation is $ 45^\circ $, and the adjacent side (distance from the boat to the base) is 100 meters. Using $ \tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}} $, we have $ 1 = \frac{h}{100} $. Solving gives $ h = 100 $ meters. Thus, the height of the lighthouse is $\boxed{100}$ meters.

["Solution: Finding the Height of a Lighthouse Using Trigonometry", "When calculating the height of a lighthouse visible from a boat, trigonometry provides a clear and powerful method. A common scenario involves the angle of elevation and the horizontal distance from the observer to the base of the structure.", "Consider the case where the angle of elevation to the top of the lighthouse is $ 45^\circ $, and the boat lies 100 meters from the base—this is a classic right triangle situation. Using the tangent function:", "$$\n\ an(\ heta) = \frac{\ ext{opposite}}{\ ext{adjacent}}\n$$", "With $ \ heta = 45^\circ $, we know $ \ an(45^\circ) = 1 $, and the adjacent side (horizontal distance) is 100 meters. Substituting values:", "$$\n1 = \frac{h}{100}\n$$", "Solving for $ h $, the height of the lighthouse:", "$$\nh = 1 \ imes 100 = 100 \ ext{ meters}\n$$", "Therefore, the lighthouse stands exactly $\boxed{100}$ meters tall.", "This straightforward application of trigonometry simplifies height estimation in navigation and engineering. By recognizing how angles and sides relate in right triangles, complex real-world problems become manageable with just basic trigonometric principles."]