A car travelled from town P to town Q on a bearing of 56 for 18km it then travel to another town R on a bearing if 200 if R is directly south of P calculate
a) Distance QR
b) Distance PR
Draw triangle PQR
QR/sin124° = PR/sin14° = 18/sin42°
PQ = 18km[56o].
P = 180-56 = 124o.
R = 200-180 = 20o.
Q = 180-124-20 = 36o.
QR/sin124 = 18/sin20
QR = 44 km.
PR/sin36 = 18/sin20
PR = 31 km.
To calculate the distances QR and PR, we can use the concept of vectors and trigonometry.
First, let's visualize the situation.
We have three towns: P, Q, and R. Town P is the starting point, and the car travels from P to Q at a bearing of 56 degrees for 18 km. Then it travels to R on a bearing of 200 degrees.
To find the distances QR and PR, we need to break down the information given into vector components.
a) Distance QR:
To calculate the distance QR, we need to find the horizontal and vertical components of the vector QR.
1. Horizontal Component (x):
We can use the trigonometric function cosine to find the horizontal component:
x = QR * cos(angle)
The angle between the north direction and the line QR is the bearing from Q to R, which is 200 degrees.
Converting this angle to radians, we have:
angle = 200 * π / 180
Now we can calculate the horizontal component:
x = QR * cos(200 * π / 180)
2. Vertical Component (y):
The vertical component can be found using the trigonometric function sine:
y = QR * sin(angle)
Using the same angle in radians, we have:
y = QR * sin(200 * π / 180)
Now we can calculate the vertical component.
To find the distance QR, we can use the Pythagorean theorem:
QR = √(x² + y²)
Substitute the values of x and y to find the distance QR.
b) Distance PR:
Since R is directly south of P, the distance between them is just the vertical component y.
Therefore, the distance PR = y.
Now, let's calculate the values using these formulas.