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 distance between Q and R correct to the nearest Km.
How do you solve it
Draw the diagram
I made a sketch, drew some verticals at Q , P and R to get the bearings entered and found angle P = 125°
so use the cosine law:
QR^2 = 50^2 + 90^2 - 2(50)(90)cos125°
...
....
QR = 125.547... or 126 to the nearest km
The distance between q and r 2. Bearing of r from p
Draw the diagram & solve it.
To solve this problem, we can use the concept of bearings and the properties of triangles.
Given that the bearing of Q from P is 075, we know that the angle between the line connecting P and Q, and the north direction is 75 degrees. Similarly, the bearing of R from P is 310, indicating that the angle between the line connecting P and R, and the north direction is 310 degrees.
Now, let's draw a diagram to represent the given information:
```
Q
/
/ 50km
P -------
\
\
R 90km
```
In the diagram, P is the common point, and the lines connecting P to Q and P to R represent distances of 50km and 90km, respectively.
To find the distance between Q and R, we need to determine the length of the line connecting Q and R in the diagram. We can do this by using the properties of triangles.
From the diagram, we can see that triangle PQR is a scalene triangle, meaning that all its sides are of different lengths.
To find the length of the side QR, we can use the Law of Cosines. According to the Law of Cosines:
c^2 = a^2 + b^2 - 2ab*cos(C)
Where "c" is the side opposite angle C, and "a" and "b" are the lengths of the other two sides.
In this case, we want to find side QR, which is opposite angle P in triangle PQR. And we know the lengths of the other two sides: PQ is 50km and PR is 90km.
So, using the Law of Cosines, we can write:
QR^2 = PQ^2 + PR^2 - 2 * PQ * PR * cos(P)
Since the bearing of Q from P is 075 (75 degrees), we can convert it to radians by multiplying it by π/180:
75 degrees * π/180 ≈ 1.3089 radians
Similarly, since the bearing of R from P is 310 (310 degrees), we can convert it to radians by multiplying it by π/180:
310 degrees * π/180 ≈ 5.4105 radians
Now, we can substitute the values into the equation:
QR^2 = 50^2 + 90^2 - 2 * 50 * 90 * cos(1.3089)
By solving this equation, we can find the value of QR.
QR ≈ 87.16 km
Therefore, the distance between Q and R, when rounded to the nearest kilometer, is approximately 87 km.