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 60° with the man's eye when at a distance of 60 metres from the tower. After 9 seconds, the angle of depression becomes 30°. The speed of the boat, assuming that it is running in still water, will be:

A man at the top of a tower watches a boat speeding away. Initially, the boat is at a distance of 60 metres from the tower and makes an angle of depression of 60° with the man's eye. After 9 seconds, the angle of depression becomes 30°.

What we need to find: The speed of the boat

Understanding the setup: Imagine you're standing on top of a tower looking down at a boat. The boat is moving away from the tower in a straight line. "Angle of depression" is the angle you look downward from horizontal to see the boat.

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When the boat is at the first position (60 m from the tower), the angle of depression is $60°$.

In a right triangle, $\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{height}}{\text{distance}}$

$\tan(60°) = \frac{h}{60}$

Since $\tan(60°) = \sqrt{3}$ (standard value to memorize):

$\sqrt{3} = \frac{h}{60}$

$h = 60\sqrt{3}$ metres

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After 9 seconds, the angle of depression becomes $30°$. Let $d$ be the new distance of the boat from the tower.

$\tan(30°) = \frac{h}{d}$

Since $\tan(30°) = \frac{1}{\sqrt{3}}$ (standard value):

$\frac{1}{\sqrt{3}} = \frac{60\sqrt{3}}{d}$

Solving for $d$:

$d = 60\sqrt{3} \times \sqrt{3}$

$d = 60 \times 3$

$d = 180$ metres

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The boat was at 60 m, now it's at 180 m from the tower.

Distance traveled $= 180 - 60 = 120$ metres

Time taken $= 9$ seconds

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$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

$\text{Speed} = \frac{120}{9}$

$\text{Speed} = \frac{40}{3}$ m/s

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Converting to km/h by multiplying with $\frac{18}{5}$ (since $1$ m/s $= 3.6$ km/h):

$\text{Speed} = \frac{40}{3} \times \frac{18}{5}$

$\text{Speed} = \frac{40 \times 18}{3 \times 5}$

$\text{Speed} = \frac{720}{15}$

$\text{Speed} = 48$ km/h

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Therefore, the speed of the boat is 48 km/h.