Question

From the top of a 100 m high cliff, the angle of depression of a boat is $45^\circ$. Calculate the distance of the boat from the foot of the cliff.

Hint:

Angle of depression = angle of elevation. Use $\tan \theta = \frac{\text{height}}{\text{base}}$ in such problems.

Answer 1

Understanding the Situation
The height of the cliff is 100 m. The angle of depression from the top of the cliff to the boat is \(45^\circ\). The angle of depression is equal to the angle of elevation from the boat to the top of the cliff. Therefore, the angle of elevation at the boat is also \(45^\circ\).

Using Trigonometric Ratio
In the right-angled triangle formed, the opposite side represents the height of the cliff and the adjacent side represents the distance of the boat from the foot of the cliff.
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \] Here,
\(\theta = 45^\circ\)
Opposite side = 100 m
Adjacent side = distance of the boat = \(x\)

Substituting the values
\[ \tan 45^\circ = \frac{100}{x} \] Since,
\(\tan 45^\circ = 1\)
\[ 1 = \frac{100}{x} \]
Solving for \(x\)
\[ x = 100 \]
Final Answer
The distance of the boat from the foot of the cliff is 100 m.