P,Q,R are three ships at sea. The bearing of Q from P is 030° and the bearing of P from R is 300°. If the distance between P and Q is 7km and that of P and R is 9km. Calculate correct to 3 significant figures the distance between.
a. Q and R
b. Bearing of Q from R
My diagram shows angle P - 90°, so
RQ^2 = 81+49
RQ = √130 = .....
For the angle at R:
tanR = 7/9
angle R = appr 37.9°
But PR was 60° from the vertical
so angle R = 22.1°
which means the bearing of Q from R is 337.9°
check my calculations
Q: Why did P, Q, and R go on a sailing adventure together?
A: They wanted to navigate the sea and get away from all their pirate problems!
To solve this question, we can use the Law of Cosines and trigonometry.
a. Distance between Q and R:
Let's label the distance between Q and R as x.
Using the Law of Cosines:
x^2 = (7km)^2 + (9km)^2 - 2(7km)(9km)cos(150°)
Calculating the cosine:
x^2 = 49km^2 + 81km^2 - 126km^2*cos(150°)
x^2 = 130km^2 - 126km^2*[-0.866] (cosine of 150° is -0.866)
x^2 = 130km^2 + 109.116km^2
x^2 = 239.116km^2
Taking the square root:
x ≈ √239.116
x ≈ 15.47 km
Therefore, the distance between Q and R is approximately 15.47 km.
b. Bearing of Q from R:
To calculate the bearing of Q from R, we need to subtract the bearing of P from R from the bearing of Q from P.
Bearing of Q from R = (Bearing of Q from P) - (Bearing of P from R)
Bearing of Q from R = 030° - 300°
Bearing of Q from R = -270°
Note: We subtract 300° from 030° since it is given that the bearing of P from R is 300°, so the bearing from R to P is -300°. Therefore, to find the bearing from R to Q, we subtract -300° from 030°, resulting in -270°.
Therefore, the bearing of Q from R is -270°.
To find the distance between Q and R, we can use the Law of Cosines. Let's label the distance between Q and R as "d".
First, let's draw a diagram to visualize the situation.
R
/|\
|
| 9km
|
P
/|\
|
| 7km
|
Q
From the given information, we know that the angle PQR is 180° - 30° - 300° = -150°. However, we need to convert this to a positive angle between 0° and 360°. To do this, we add 360°: -150° + 360° = 210°.
Now, we can use the Law of Cosines:
d^2 = 7^2 + 9^2 - 2(7)(9)cos(210°)
Calculating this:
d^2 = 49 + 81 - 126cos(210°)
d^2 = 130 - 126(-0.866)
d^2 = 130 + 109.116
d^2 = 239.116
Taking the square root of both sides, we get:
d ≈ √(239.116)
d ≈ 15.464
Therefore, the distance between Q and R is approximately 15.464 km.
To find the bearing of Q from R, we can use the formula:
Bearing of Q from R = 180° - (Bearing of R from P + 180° - Bearing of Q from P)
Using the given bearings, we can substitute them into the formula:
Bearing of Q from R = 180° - (300° + 180° - 30°)
Bearing of Q from R = 180° - 300° - 180° + 30°
Bearing of Q from R = -270° + 30°
Bearing of Q from R = -240°
Again, we need to convert this to a positive angle between 0° and 360°. Adding 360°:
Bearing of Q from R = -240° + 360°
Bearing of Q from R = 120°
Therefore, the bearing of Q from R is 120°.