Watch Tower Height and Depression Angles

Let the height of the watch tower be denoted by h. We are given h = 200 m.

Let the base of the watch tower be point C, and the top be point D. Let the two cliffs be points A and B, located in a horizontal line through C on the same side.

The angles of depression from the top of the tower (D) to the cliffs are $45^{\circ}$ and $30^{\circ}$. Since the cliffs are on the same side and in a line with the base, the cliff with the larger angle of depression is closer to the tower.

• Angle of depression to the nearer cliff (A) = $45^{\circ}$.
• Angle of depression to the farther cliff (B) = $30^{\circ}$.

These angles of depression correspond to the angles of elevation from the cliffs to the top of the tower.

Calculating Distances from Tower Base

Consider the right-angled triangle formed by the tower and the nearer cliff (ADC).

• The angle of elevation from A to D is $45^{\circ}$.
• Using trigonometry, $\tan(45^{\circ}) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{DC}{AC} = \frac{h}{AC}$.
• Therefore, $AC = \frac{h}{\tan(45^{\circ})} = \frac{200 \text{ m}}{1} = 200 \text{ m}$.

Consider the right-angled triangle formed by the tower and the farther cliff (BDC).

• The angle of elevation from B to D is $30^{\circ}$.
• Using trigonometry, $\tan(30^{\circ}) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{DC}{BC} = \frac{h}{BC}$.
• Therefore, $BC = \frac{h}{\tan(30^{\circ})} = \frac{200 \text{ m}}{1/\sqrt{3}} = 200\sqrt{3} \text{ m}$.

Finding Distance Between Cliffs

Since the cliffs A and B are on the same side of the tower and in a line through the base C, the distance between them is the difference between their distances from the base.

• Distance AB = BC - AC
• Distance AB = $200\sqrt{3} \text{ m} - 200 \text{ m}$
• Distance AB = $200(\sqrt{3} - 1) \text{ m}$

This matches option 3.