In this problem, you can use trigonometry to find the height of the cliff. The angle of depression from point A (top of the cliff) to point B (the ship) is $10^\circ$, and the horizontal distance from point B to the foot of the cliff at point X is 800 meters.

Here’s the setup:

• The angle of depression ($\theta$) is $10^\circ$.
• The distance $BX$ (the horizontal distance) is 800 meters.
• Let $h$ represent the height of the cliff, which is what we need to find.

Step-by-step solution:

Since the angle of depression is measured from a horizontal line at the top of the cliff, the angle between the height of the cliff and the line of sight from point A to B (in triangle AXB) is also $10^\circ$. This means you can use the tangent of the angle to relate the height of the cliff and the horizontal distance.

Using the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{BX}$

Substitute the known values: $\tan(10^\circ) = \frac{h}{800}$

Now, solve for $h$: $h = 800 \times \tan(10^\circ)$

Using a calculator: $h = 800 \times 0.1763 \quad (\text{since } \tan(10^\circ) \approx 0.1763)$ $h \approx 141.04 \, \text{meters}$

Final answer:

The height of the cliff is approximately 141 meters, correct to the nearest metre.

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Would you like to know more about the trigonometric relationships or further clarification on this process?

Here are five related questions:

1. How do you calculate the horizontal distance if you are given the height and the angle of depression?
2. What is the difference between the angle of elevation and the angle of depression?
3. How can you use the sine or cosine functions in similar problems?
4. What is the formula for calculating distances in a right triangle using angles?
5. Can the angle of depression be more than 90 degrees in some cases?

Tip: Always remember that the angle of depression from a horizontal line is equal to the angle of elevation from the point being observed due to alternate angles in geometry.