Two ships are sailing in the sea on the two sides of a lighthouse. The angles of elevation of the top of the lighthouse as observed from the two ships are $30^\circ$ and $45^\circ$ respectively. If the lighthouse is 100 m high, the distance between the two ships is

### Concept & Opposite Sides Addition When two points are on opposite sides of a central vertical object, the total distance between them is the sum of their individual horizontal distances from the object's base. $$ \text{Total Distance} = d_1 + d_2 = \frac{h}{\tan(\theta_1)} + \frac{h}{\tan(\theta_2)} $$ ### Step-by-Step Solution 1. Given: Height of lighthouse ($h$) = $100$ m. Angles of elevation are $30^\circ$ and $45^\circ$. 2. Let $d_1$ be the distance of the first ship (at $30^\circ$). $$ \tan(30^\circ) = \frac{100}{d_1} \Rightarrow \frac{1}{\sqrt{3}} = \frac{100}{d_1} \Rightarrow d_1 = 100\sqrt{3} $$ 3. Let $d_2$ be the distance of the second ship (at $45^\circ$). $$ \tan(45^\circ) = \frac{100}{d_2} \Rightarrow 1 = \frac{100}{d_2} \Rightarrow d_2 = 100 $$ 4. Calculate the total distance between the ships: $$ \text{Distance} = d_1 + d_2 = 100\sqrt{3} + 100 $$ 5. Substitute $\sqrt{3} \approx 1.732$: $$ 100(1.732) + 100 = 173.2 + 100 = 273.2 \text{ m} $$ 6. The closest option is $273$ m. ### Exam Strategy & Shortcut A $45^\circ$ angle means the distance equals the height ($100$). A $30^\circ$ angle means the distance is height $\times \sqrt{3}$ ($173.2$). Since they are on opposite sides, simply add them: $100 + 173.2 = 273.2$. ### Common Pitfall Subtracting the distances instead of adding them. Subtraction is only used when both objects are on the *same side* of the lighthouse. ### Final Answer Therefore, the correct answer is **273 m**.