3. The angle of depression of an aeroplane measured from a control tower, PQ, of height 88.9 m is 48°. When the plane moves along the runway from point Ato point Band stops, the angle of depression becomes 25.2°. The distance from point P to point Ais given as 119.6 m. Complete the diagram below to represent this informatior Р brid (ii) Leaving your answers correct to ONE decimal place calculate (a) Durmine the distance from the control tower to point A. (2) (b) Calculate the distance moved by the plane from its initial position.

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ConnollyD60783 ConnollyD60783 · Answer 1 · 2022-11-03T13:05:37+01:00

The Solution:

Part (a)

Representing the problem fully in a diagram, we have:

Part (b)

We are required to find the length of QA= x.

We shall use Trigonometrical Ratio as below:

[tex]\begin{gathered} tan48^o=\frac{opposite}{adjacent}=\frac{88.9}{x} \\ \\ tan48=\frac{88.9}{x} \end{gathered}[/tex]

Making x the subject of the formula, we get

[tex]x=\frac{88.9}{tan48}=80.0459\approx80.0m[/tex]

Thus, the distance from the control tower to point A is 80.0 meters.

Part (c)

We are required to find the length of AB= y.

Considering triangle PQB, we have:

[tex]\begin{gathered} tan25.2=\frac{88.9}{x+y}=\frac{88.9}{80+y} \\ \\ 0.47056=\frac{88.9}{80+y} \end{gathered}[/tex]

Solving for y, we get

[tex]\begin{gathered} 80+y=\frac{88.9}{0.47056} \\ \\ y=188.922-80 \\ y=108.922\approx108.9m \end{gathered}[/tex]

Thus, the distance moved by the plane from its initial position is 108.9 meters.