Heights and Distance

DEFINITIONS

Angle of elevation:

If 'O' be the observer's eye and OX be the horizontal line through O. If the object P is at a higher level than eye, then angle POX = $\theta$ is called the angle of elevation.

Angle of Depression:

If the object P is at a lower level than O, then angle POX is called the angle of depression.

Illustration 1: A boat is being rowed away from a cliff 150 meter height. At the top of the cliff the angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat.

Solution Let PQ be the cliff $\Rightarrow$ PQ = 150 meters.

Let the two position of the boat be at A and B so that AP = h cot60 = 150/Ö3

BP = 150 cot45° = 150

$\Rightarrow$ AB = 150 - $\frac{150}{\sqrt{3}}=\frac{150}{\sqrt{3}}$$\left(\sqrt{3};-1\right)$The speed of the boat is $\frac{1}{2}$.$\frac{150}{\sqrt{3}}$$\left(\sqrt{3}-1\right)$ x 60 metres per hour.

= $\frac{4500}{\sqrt{3}}$ $\left(\sqrt{3}-1\right)$ meters per hour.