Three towns p, q and r are such that the distance between p and q is 50km and the distance between p and r is 90km. If the bearing of q from p is 075 and the bearing of r from p is 310, find the
(a)distance between q and r
(b)bearing of r from q
answered above
A)distance qr=
|ar|^2=q^2+r^2-2pqcosP
=90^2+50^2-2*50*90cos125
=8100+2500-9000cos125
=10600+5162.1879
=root of 10600+5162.1879
qr=130
b)The bearing of r from q=
=q/sinQ=p/sinP
=90/sinQ=130/sin125
=130sinQ=90sin125
Q(sin)=73.72/130
Q=34.6
Then the bearing of r from q=360-34.6
=325.4
To find the distance between towns q and r, we can use the Law of Cosines:
(a) distance between q and r:
Using the Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C)
Let PQ be the distance between p and q (which is given as 50 km)
Let PR be the distance between p and r (which is given as 90 km)
Let QR be the distance between q and r.
From the given information, we have:
PQ = 50 km
PR = 90 km
angle PQR = 310 - 75 = 235 degrees (converted the bearing to the angle)
Using the Law of Cosines, we can calculate QR:
QR^2 = PQ^2 + PR^2 - 2 * PQ * PR * cos(PQR)
QR^2 = 50^2 + 90^2 - 2 * 50 * 90 * cos(235 degrees)
QR^2 = 2500 + 8100 - 9000 * (-0.2588) (converted cos(235 degrees) to decimal)
QR^2 = 10640 + 2329.2
QR^2 = 12969.2
QR = sqrt(12969.2)
QR ≈ 113.88 km
Therefore, the distance between towns q and r is approximately 113.88 km.
(b) bearing of r from q:
To determine the bearing, we can use basic trigonometry.
Let θ be the bearing of r from q.
From the given information, we have:
angle PQR = 235 degrees (as calculated in part (a))
To find the bearing of r from q, we need to calculate the angle PQR relative to the bearing of q from p.
The angle we need to calculate is the supplementary angle to the angle PQR. In other words, the bearing of r from q = 235° (180° - PQR).
Therefore, the bearing of r from q is approximately 180° - 235° = -55°.
Note: In the context of bearings, a negative angle represents a bearing in the opposite direction, so we can also write the bearing of r from q as 305° (55° in the opposite direction).
Hence, the bearing of r from q is approximately 305°.
To solve this problem, we can use the concept of bearings and trigonometry. We will use the given information to draw a diagram and then apply trigonometric formulas.
(a) Distance between q and r:
Let's draw a diagram representing the three towns P, Q, and R. Choose a scale that is convenient, for example, one unit on the diagram can represent 10km.
1. Start by labeling the points P, Q, and R on the diagram.
2. From the information given, we know that the distance between P and Q is 50km. So, draw a line segment PQ with a length of 5 units on the diagram.
3. The bearing of Q from P is given as 075 (assuming it's measured clockwise from the North direction). This means that the angle between the line segment PQ and the North direction is 75 degrees. Draw an angle of 75 degrees at point P, opening towards the East direction.
4. The bearing of R from P is given as 310. This means that the angle between the line segment PR and the North direction is 310 degrees, measured clockwise. Draw an angle of 310 degrees at point P, opening towards the West direction.
5. The distance between P and R is given as 90km. So, draw a line segment PR with a length of 9 units on the diagram.
6. Now, to find the distance between Q and R, we can use the Law of Cosines:
distance^2 = PQ^2 + PR^2 - 2 x PQ x PR x cos(angle between PQ and PR)
angle between PQ and PR = 310 - 75 = 235 degrees (angle at P between QP and RP)
distance^2 = 5^2 + 9^2 - 2 x 5 x 9 x cos(235)
Calculate the value of distance to find the actual distance between Q and R.
(b) Bearing of R from Q:
To find the bearing of R from Q, we can use the concept of subtraction of bearings.
1. The bearing of Q from P is given as 075.
2. The bearing of R from P is given as 310.
3. To find the bearing of R from Q, we subtract the bearing of Q from P from the bearing of R from P:
Bearing of R from Q = Bearing of R from P - Bearing of Q from P
Bearing of R from Q = 310 - 75
Calculate the value to find the bearing of R from Q.