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

Concept: The angle of depression is equal to the angle of elevation. Using right triangle trigonometry: \[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \]
Step 1: Interpret the problem. Height of cliff = 100 m Angle of depression = $45^\circ$ So angle of elevation from the boat = $45^\circ$. Let distance from foot of cliff = $x$ m.
Step 2: Apply tangent ratio. \[ \tan 45^\circ = \frac{\text{height}}{\text{distance}} \] \[ 1 = \frac{100}{x} \]
Step 3: Solve for $x$. \[ x = 100 \text{ m} \]
Conclusion: The distance of the boat from the foot of the cliff is 100 m.