A search-and-rescue helicopter leaves a central station P and flies 16\text{ km} on a bear… — JAMB UTME

A search-and-rescue helicopter leaves a central station $P$ and flies $16\text{ km}$ on a bearing of $050^\circ$ to reach a waypoint $Q$ . It then changes course and flies $12\text{ km}$ on a bearing of $140^\circ$ to reach a mountain rescue site $R$ . From the central station $P$ , the angle of elevation to the helicopter hovering vertically above point $R$ is $45^\circ$ . What is the vertical height of the helicopter above the horizontal plane of station $P$ , in kilometers?

Answer: 20 km

Answer

The vertical height of the helicopter above the horizontal plane of station

P

20\text{ km}

The horizontal journey forms a right-angled triangle

PQR

with side lengths

16\text{ km}

and

12\text{ km}

, yielding a hypotenuse (horizontal distance

PR

) of

20\text{ km}

. Since the angle of elevation from

P

to the hovering helicopter is

45^\circ

, the vertical height is equal to

20 \times \tan(45^\circ) = 20\text{ km}

Step-by-Step Solution

Find the back bearing of station

P

from waypoint

Q

Back bearing =

050^\circ + 180^\circ = 230^\circ

To determine the enclosed interior angle at point

Q

, the reverse direction from

Q

P

must be calculated.

Calculate the interior angle

\angle PQR

\angle PQR = 230^\circ - 140^\circ = 90^\circ

Subtracting the forward bearing of

R

from the back bearing of

P

yields the right angle between the two paths.

Calculate the horizontal displacement distance

PR

PR = \sqrt{16^2 + 12^2} = 20\text{ km}

Since

\triangle PQR

is right-angled at

Q

, the distance

PR

is obtained using the Pythagorean theorem.

Determine the vertical altitude using trigonometry

\text{Height} = PR \times \tan(45^\circ) = 20 \times 1 = 20\text{ km}

In the vertical right-angled triangle formed by

P

, the ground projection of

R

, and the helicopter,

\tan(45^\circ) = \frac{\text{Height}}{\text{Horizontal Distance}}

Key Concept

Combining 3-point bearings in 2D with right-triangle trigonometry for 3D angles of elevation.

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