Thus, \( h(-1) = \boxed{0} \).Question: An oceanographer measures the angle of elevation to the top of a lighthouse from a boat as $ 45^\circ $. If the boat is 100 meters from the base of the lighthouse, what is the height of the lighthouse?
["Understanding the Science Behind the Lighthouse Height: A Problem in Trigonometry", "When an oceanographer measures the angle of elevation to the top of a lighthouse from a boat, they apply fundamental trigonometric principles. This real-world scenario illustrates one of the most practical uses of right triangle geometry — determining heights of distant or inaccessible objects using angles and known distances.", "In this case, the situation is straightforward: from a boat, the angle of elevation to the lighthouse’s peak is measured at ( 45^\circ ), and the boat lies exactly 100 meters from the base of the lighthouse. What is the height of the lighthouse?", "---", "### The Mathematical Foundation", "The angle of elevation from the observer’s line of sight to the top of the lighthouse is ( 45^\circ ). Using trigonometry in a right triangle — formed by the boat, the base of the lighthouse, and the lighthouse’s top — we use the tangent function, defined as the ratio of the opposite side (lighthouse height ( h )) to the adjacent side (horizontal distance):", "[\n\ an(\ heta) = \frac{\ ext{opposite}}{\ ext{adjacent}} = \frac{h}{d}\n]", "Given:\n- ( \ heta = 45^\circ )\n- ( d = 100 ) meters", "Substituting values:", "[\n\ an(45^\circ) = \frac{h}{100}\n]", "We know from trigonometric identities that:", "[\n\ an(45^\circ) = 1\n]", "Thus:", "[\n1 = \frac{h}{100}\n]", "Multiplying both sides by 100 gives:", "[\nh = 100\n]", "Therefore, the height of the lighthouse is 100 meters.", "---", "### Why Is the Height Exactly 100 Meters?", "This result stems from the unique behavior of the tangent function at ( 45^\circ ): the rise equals the run, meaning for every meter you move horizontally, the vertical height increases by the same amount. Because the distance from the boat is exactly 100 meters and the angle is ( 45^\circ ), the height matches the distance — a rare but elegant coincidence in trigonometry.", "---", "### Real-World Application: Trigonometry Beyond the Classroom", "Oceanographers and surveyors routinely use such angular measurements to map coastlines, estimate structures, and navigate safely without direct access. This problem exemplifies how mathematical principles enable accurate measurements across vast distances, transforming uncertainty into certainty.", "---", "### Final Answer", "Thus, ( h(-1) = \boxed{0} ) — though in this scenario, ( h = 100 ) meters — but more importantly, this example reinforces how ( \ an(45^\circ) = 1 ) leads to a clean, exact solution. When the angle of elevation is ( 45^\circ ) and the horizontal distance is 100 meters, the lighthouse’s height is precisely equal to that distance.", "---", "Key Takeaway: Always remember that ( \ an(45^\circ) = 1 ), so any horizontal distance ( d ) yields a height ( h = d ) when the angle of elevation is ( 45^\circ ). This fundamental relationship simplifies many real-world height calculations — a perfect fusion of theory and practice."]
