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Difficulty: Very hardAngles of Elevation, Depression, and Bearings

A boat sails 10 km10\text{ km} from a port PP on a bearing of 040∘040^\circ to a point QQ. From QQ, it changes direction and sails 103 km10\sqrt{3}\text{ km} on a bearing of 130∘130^\circ to reach a point RR. What is the bearing of port PP from point RR?

  1. 280∘280^\circAnswer
  2. B
    100∘100^\circ
  3. C
    250∘250^\circ
  4. D
    160∘160^\circ

Answer

The bearing of port P from point R is 280∘280^\circ.
By drawing North reference lines at P, Q, and R, the back bearing of P from Q is 040∘+180∘=220∘040^\circ + 180^\circ = 220^\circ. The bearing of R from Q is 130∘130^\circ, creating an interior right angle of 220∘−130∘=90∘220^\circ - 130^\circ = 90^\circ at Q. Using right-triangle trigonometry, tan⁡(∠QPR)=10310=3\tan(\angle QPR) = \frac{10\sqrt{3}}{10} = \sqrt{3}, giving ∠QPR=60∘\angle QPR = 60^\circ. Adding this to the initial bearing of 040∘040^\circ gives the bearing of R from P as 100∘100^\circ. Finally, adding 180∘180^\circ gives the reverse bearing of P from R as 280∘280^\circ.

Step-by-Step Solution

1
Determine the interior angle ∠PQR\angle PQR at vertex QQ
∠PQR=90∘\angle PQR = 90^\circ
The back bearing of P from Q is 040∘+180∘=220∘040^\circ + 180^\circ = 220^\circ. The bearing of R from Q is 130∘130^\circ. The interior angle between QP and QR is 220∘−130∘=90∘220^\circ - 130^\circ = 90^\circ.
2
Calculate the angle ∠QPR\angle QPR inside right-angled triangle PQRPQR
∠QPR=60∘\angle QPR = 60^\circ
Since triangle PQRPQR is right-angled at QQ, tan⁡(∠QPR)=oppositeadjacent=QRPQ=10310=3\tan(\angle QPR) = \frac{\text{opposite}}{\text{adjacent}} = \frac{QR}{PQ} = \frac{10\sqrt{3}}{10} = \sqrt{3}. Therefore, ∠QPR=arctan⁡(3)=60∘\angle QPR = \arctan(\sqrt{3}) = 60^\circ.
3
Find the forward bearing of point RR from port PP
Bearing of RR from P=100∘P = 100^\circ
The bearing of Q from P is 040∘040^\circ. Since R lies to the right (clockwise) of segment PQ, add ∠QPR=60∘\angle QPR = 60^\circ to 040∘040^\circ: 040∘+60∘=100∘040^\circ + 60^\circ = 100^\circ.
4
Compute the back bearing of port PP from point RR
Bearing of PP from R=280∘R = 280^\circ
The back bearing is obtained by adding 180∘180^\circ to the forward bearing from P to R: 100∘+180∘=280∘100^\circ + 180^\circ = 280^\circ.

Key Concept

Three-point bearing calculations using right-triangle trigonometry and back bearings
Estimated Time:2m 30s
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Topics

Angles of Elevation, Depression, and Bearings

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