A man is watching from the top of a tower a boat speeding away from the tower. The boat makes an angle of depression of $45^\circ$ with the man's eye when at a distance of 60 metres from the tower. After 5 seconds, the angle of depression becomes $30^\circ$. What is the approximate speed of the boat, assuming that it is running in still water?

### Concept & Trigonometry and Speed The speed of an object is the distance it travels divided by the time taken. Here, we must first use trigonometry to find the height of the tower and then the additional distance the boat covers when the angle of depression changes. $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$ ### Step-by-Step Solution 1. Let the height of the tower be $h$. 2. Initial position: The boat is at an angle of depression of $45^\circ$ (which means the angle of elevation from the boat is also $45^\circ$) and a distance of $60$ m. $$ \tan(45^\circ) = \frac{h}{60} \Rightarrow 1 = \frac{h}{60} \Rightarrow h = 60 \text{ m} $$ 3. Final position: After $5$ seconds, the angle of depression becomes $30^\circ$. Let the new total distance from the tower be $d$. $$ \tan(30^\circ) = \frac{h}{d} \Rightarrow \frac{1}{\sqrt{3}} = \frac{60}{d} \Rightarrow d = 60\sqrt{3} \text{ m} $$ 4. The distance traveled by the boat in $5$ seconds is: $$ \text{Distance} = d - 60 = 60\sqrt{3} - 60 = 60(\sqrt{3} - 1) \text{ m} $$ 5. Substitute $\sqrt{3} \approx 1.732$: $$ \text{Distance} = 60(1.732 - 1) = 60(0.732) = 43.92 \text{ m} $$ 6. Calculate speed in m/s: $$ \text{Speed} = \frac{43.92}{5} = 8.784 \text{ m/s} $$ 7. Convert speed from m/s to kmph by multiplying by $\frac{18}{5}$ (or $3.6$): $$ \text{Speed in kmph} = 8.784 \times 3.6 = 31.6224 \text{ kmph} $$ 8. The approximate speed is $32$ kmph. ### Exam Strategy & Shortcut For angles changing from $45^\circ$ to $30^\circ$, the extra distance traveled is always $h(\sqrt{3} - 1)$. Since $h=60$, distance is $60(\sqrt{3}-1)$. Just evaluate $60 \times 0.732 / 5 \times 3.6 \approx 31.6$. Round to $32$. ### Common Pitfall Forgetting to convert the final speed from metres per second (m/s) to kilometres per hour (kmph), which would lead to looking for an answer around $8.8$ and guessing incorrectly. ### Final Answer Therefore, the correct answer is **32 kmph**.