Question

1. a search and rescue helicopter is flying towards a ship in distress. the helicopter is at an altitude (height) of 1200 metres, and the angle of depression from the helicopter to the front of the ship at water level is 18°.

(a) label the angle of depression and the altitude of the helicopter on the diagram below. diagram of ship and helicopter
(b) hence, find the horizontal distance between the front of the ship and the water directly below the helicopter. hand-drawn triangle with labels, formula: $sin 18^circ = \frac{x}{1200}$, calculation steps

Explanation:

Part (a)

To label the angle of depression and the altitude:

• Altitude: The vertical distance from the helicopter to the water level is the altitude, so we label the vertical segment (from the helicopter to the water) as \( 1200 \) metres.
• Angle of Depression: The angle of depression is the angle between the horizontal line from the helicopter and the line of sight to the ship. So we label the angle between the horizontal (from the helicopter) and the line connecting the helicopter to the ship's front as \( 18^\circ \).

Part (b)

Step 1: Identify the trigonometric relationship

We have a right - triangle where the opposite side to the angle of depression (when considering the alternate interior angle, which is equal to the angle of depression) is the altitude (\( 1200 \) m) and the adjacent side is the horizontal distance (\( x \)) we want to find. The angle of depression is \( 18^\circ \), and we can use the tangent function. Recall that \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \). Here, \( \theta = 18^\circ \), opposite \( = 1200 \) m, and adjacent \( = x \). So \( \tan(18^\circ)=\frac{1200}{x} \).

Step 2: Solve for \( x \)

From \( \tan(18^\circ)=\frac{1200}{x} \), we can re - arrange the formula to solve for \( x \). Cross - multiplying gives us \( x=\frac{1200}{\tan(18^\circ)} \). We know that \( \tan(18^\circ)\approx0.3249 \). Then \( x = \frac{1200}{0.3249}\approx3693 \) metres (the previous attempt in the image used the wrong trigonometric function; sine was used instead of tangent. The correct approach is to use tangent because we have the opposite and adjacent sides in the right - triangle for the angle of depression's alternate interior angle).

Answer:

• Part (a): Label the vertical side (from helicopter to water) as \( 1200 \) m (altitude) and the angle between the horizontal from the helicopter and the line of sight to the ship as \( 18^\circ \) (angle of depression).
• Part (b): The horizontal distance is approximately \( \boldsymbol{3693} \) metres (or more precisely, using a calculator for \( \tan(18^\circ) \): \( \tan(18^\circ)\approx0.3249 \), \( x=\frac{1200}{0.3249}\approx3693 \) m).