1. **Multiple Choice Questions - Answers:** 1. The angle measured clockwise from north is called a **bearing**. Answer: c. Bearings 2. If a ship sailed south then west, the resulting direction is **southwest**. Answer: d. SW 3. The angle formed from a location above ground to a point below is the **angle of depression**. Answer: d. Depression 4. When measuring bearing, the starting point is always **north**. Answer: a. North 5. If the bearing of P from Q is 30°, then the bearing of Q from P is $30^\circ + 180^\circ = 210^\circ$. Answer: a. 210 2. **Structured Question** **Problem statement:** A man 1.6m tall stands on top of a cliff 56.4m high and observes two ships in the same direction. The angle of depression to ship A is $30^\circ$ and to ship B is $40^\circ$. Find the distance between ships A and B. **Step 1: Define heights and points** - Total height of man’s eyes from ground: $56.4 + 1.6 = 58.0$ m - Let the distance from the cliff base to ship A be $d_A$ and to ship B be $d_B$. **Step 2: Use angle of depression and right-angle triangle geometry** - Angle of depression corresponds to angle between horizontal and line of sight downward. - Using right triangle from man’s eye to ship, distance is adjacent side, height is opposite side. For ship A: $$\tan 30^\circ = \frac{58}{d_A} \implies d_A = \frac{58}{\tan 30^\circ}$$ For ship B: $$\tan 40^\circ = \frac{58}{d_B} \implies d_B = \frac{58}{\tan 40^\circ}$$ **Step 3: Calculate distances $d_A$ and $d_B$** $$\tan 30^\circ = \frac{\sqrt{3}}{3} \approx 0.5774$$ $$d_A = \frac{58}{0.5774} \approx 100.48 \text{ m}$$ $$\tan 40^\circ \approx 0.8391$$ $$d_B = \frac{58}{0.8391} \approx 69.11 \text{ m}$$ **Step 4: Determine distance between ships A and B** Since ships A and B lie in the same direction from the cliff, distance between them is: $$|d_A - d_B| = |100.48 - 69.11| = 31.37 \text{ m}$$ **Final answer:** The distance between ships A and B is approximately **31.37 meters**. **Slug:** bearings distance **Subject:** trigonometry