Question

From the top of a cliff 126 meters high, the angle of depression of a boat is 20.7 degrees. Hwo far is the boat from the foot of the cliff?

Answer 1

Answer on Question #71010 – Math – Trigonometry

Question

From the top of a cliff 126 meters high, the angle of depression of a boat is 20.7 degrees. How far is the boat from the foot of the cliff?

Solution

The angle of depression is the angle $\beta$ measured downward from the horizontal line in the direction of sight of a boat (see figure).

By condition of the problem $\beta = 20.7{}^{\circ}$. Then we can find the angle $\alpha$ between the cliff, AC, and the line of sight of a boat, CB:

$$\alpha = 90{}^{\circ} - \beta = 90{}^{\circ} - 20.7{}^{\circ} = 69.7{}^{\circ}$$

Now we consider $\Delta ABC$ which is the right triangle since $m \angle CAB = 90{}^{\circ}$. The tangent ratio for an acute angle $\alpha$ in this triangle is

$$\tan \alpha = \frac{AB}{AC}$$

Then we get the distance from the foot of the cliff A to the boat B, that is AB:

$$AB = AC \cdot \tan \alpha$$

Substituting $AC = 126 \, \text{m}$ and $\tan \alpha = \tan 69.7{}^{\circ} \approx 2.7$ we get

$$AB = 126 \cdot 2.7 \approx 340 \, \text{m}$$

Answer: the distance from the foot of the cliff to the boat is $340 \, \text{m}$.

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