The bearing of Q from P is150 and the bearing of P from R is 015. If Q and R are 24km and 32km respectively from P. Represent this information into a diagram. Calculate the distance between Qand R. Find the bearing of R from Q.
I hope you made your sketch.
by vectors
vector RQ = (32cos15°, 32sin15°) + (24cos150, 24sin150°)
= (30.9096..., 8.2822...) + (-20.7846.., 12)
= (10.125.. , 20.2822...)
maginitude = √(10.125..^2 + 20.2822...^2) = appr 22.669
or
using the cosine law:
my diagram had a triangle of sides 32 and 24 with an angle of 45 ° between them
PR^2 = 24^2 + 32^2 - 2(24)(32)cos45°
= 576 + 1024 - 1086.116..
= 513.8839...
PR = √513.8839... = appr 22.669 , just like above
All angles are measured CW from +y-axis.
Given: PQ = 24km[150o], RP = 32km[15o].
a. RQ = RP + PQ = 32[15o] + 24[150o]
RQ = (32*sin15+24*sin150) + (32*cos15+24*cos150)I
RQ = 20.3 + 10.1i. = 22.7km[63.5o].
b. QR = 22.7[63.5+180] = 22.7km[243.5o].
Well, let's start by representing the information into a diagram.
Assume that P is the starting point, Q is located 150 degrees to the right of the North direction, and R is located 15 degrees to the left of the North direction.
R(32km)
|
|
|\
015| \ 150
| \
| \
| \
|-----P(24km)
Q(24km)
Now, to calculate the distance between Q and R, we can use the Pythagorean theorem since we have a right triangle. The distance between Q and R will be the hypotenuse of the triangle formed by P, Q, and R.
Using the Pythagorean theorem, we can calculate the distance between Q and R:
Distance^2 = (Distance between Q and P)^2 + (Distance between R and P)^2
Distance^2 = (24km)^2 + (32km)^2
Distance^2 = 576km^2 + 1024km^2
Distance^2 = 1600km^2
Distance = √1600km^2
Distance = 40km
So, the distance between Q and R is 40km.
Now, let's find the bearing of R from Q. Since we know that Q is 150 degrees to the right of the North direction, we can subtract 150 degrees from it to find the bearing of R from Q.
Bearing of R from Q = Bearing of Q from North - Angle QPR
Bearing of R from Q = 150 degrees - 15 degrees
Bearing of R from Q = 135 degrees
Therefore, the bearing of R from Q is 135 degrees.
To represent the information in a diagram, draw three points labeled P, Q, and R. Place R 32km away from P, and Q 24km away from P. Label the angle between the line segments PR and PQ as 150°. Also, label the angle between the line segments QR and QP as 15°.
To calculate the distance between Q and R, you can use the cosine rule. According to the cosine rule, the square of the side opposite an angle in a triangle is equal to the sum of the squares of the other two sides minus twice the product of those two sides multiplied by the cosine of the angle between them. In our case, we can use the triangle PQR to calculate the distance QR.
Using the cosine rule:
QR^2 = QP^2 + RP^2 - 2(QP)(RP)cos(150)
Substituting the given values:
QR^2 = 24^2 + 32^2 - 2(24)(32)cos(150)
QR^2 = 576 + 1024 - 1536(cos150)
QR^2 = 1600 - 1536(-0.866)
QR^2 = 1600 + 1330.176
QR^2 ≈ 2930.176
Taking the square root of both sides:
QR ≈ √2930.176
QR ≈ 54.09 km
Therefore, the distance between Q and R is approximately 54.09 km.
To find the bearing of R from Q, subtract the bearing of R from P (015) from the bearing of Q from P (150).
Bearing of R from Q = 150 - 15
Bearing of R from Q = 135
Therefore, the bearing of R from Q is 135°.
To represent this information into a diagram, you can use a compass rose. Draw a straight line to represent each of the locations: P, Q, and R. Label each line with its respective distance, as given in the question (24km for Q and 32km for R).
To find the distance between Q and R, you can use the Pythagorean theorem. The distance between two points can be calculated as the square root of the sum of the squares of the differences in their x and y coordinates. In this case, the distances are the legs of a right triangle, and the hypotenuse will represent the distance between Q and R.
Let's calculate it step by step:
1. Calculate the differences in the x and y coordinates between Q and R:
- x-coordinate difference: 24km - 0km = 24km
- y-coordinate difference: 32km - 0km = 32km
2. Apply the Pythagorean theorem:
Distance between Q and R = √(24km^2 + 32km^2)
Distance between Q and R = √(576km^2 + 1024km^2)
Distance between Q and R = √(1600km^2)
Distance between Q and R = 40km
So, the distance between Q and R is 40km.
To find the bearing of R from Q, we can use the angle addition rule. The bearing of Q from P is given as 150°, and the bearing of P from R is given as 015°.
1. Subtract the bearing of P from R from the bearing of Q from P:
150° - 015° = 135°
So, the bearing of R from Q is 135°.