Question

e2. from the top of a cliff, the angles of depression to two boats aligned along the same line of sight are 12° and 20°. the boats are 240 m apart. assuming constant sea level, find the height of the cliff (nearest meter). hint: create a non - right triangle using lines of sight.

Explanation:

Step1: Find the angles of the non - right triangle

The angle of depression is equal to the angle of elevation. Let the height of the cliff be \(h\).
The angle between the two lines of sight in the non - right triangle is \(\theta=20^{\circ}- 12^{\circ}=8^{\circ}\).
The angle opposite to the side of length \(240\) m in the non - right triangle (using the angle of elevation concept) is \(180^{\circ}-20^{\circ}=160^{\circ}\), and the other non - right triangle angle (adjacent to the longer line of sight) is \(12^{\circ}\).

Step2: Use the sine rule

Let the length of the line of sight with an angle of elevation of \(12^{\circ}\) be \(x\). By the sine rule \(\frac{x}{\sin160^{\circ}}=\frac{240}{\sin8^{\circ}}\).

$$x = \frac{240\times\sin160^{\circ}}{\sin8^{\circ}}$$

Since \(\sin160^{\circ}=\sin(180^{\circ} - 20^{\circ})=\sin20^{\circ}\), then \(x=\frac{240\times\sin20^{\circ}}{\sin8^{\circ}}\)

Step3: Relate to the height of the cliff

We know that \(\sin12^{\circ}=\frac{h}{x}\), so \(h = x\sin12^{\circ}\)
Substitute \(x=\frac{240\times\sin20^{\circ}}{\sin8^{\circ}}\) into \(h = x\sin12^{\circ}\)

$$h=\frac{240\times\sin20^{\circ}\times\sin12^{\circ}}{\sin8^{\circ}}$$

We know that \(\sin20^{\circ}\approx0.3420\), \(\sin12^{\circ}\approx0.2079\), \(\sin8^{\circ}\approx0.1392\)

$$h=\frac{240\times0.3420\times0.2079}{0.1392}$$

$$h=\frac{240\times0.0711}{0.1392}$$

$$h=\frac{17.064}{0.1392}\approx123$$

Answer:

\(123\) m