from a point R,300m north of P,a man walks east wards to a place Q which is 600m from P find the bearing oF P from Q,correct to the nearest degree and the distance between R and Q.

Question

from a point R,300m north of P,a man walks east wards to a place Q which is 600m from P find the bearing oF P from Q,correct to the nearest degree and the distance between R and Q.

Answer 1

sin PQR = 300/600 = 1/2

so
PQR = 30 degrees
that is 30 degrees south of west
or 60 degrees west of south
which id a compass bearing of
180 + 60
= 240 degrees clockwise from north

cos 30 = RQ/600
RQ = 520 m

Answer 2

To find the bearing of point P from point Q, we can use the formula:

Bearing = arctan(AD / AQ)

where AD is the eastward distance and AQ is the northward distance.

Given that the man walks eastwards from point R to point Q, the eastward distance, AD, is equal to 600 m.

The northward distance, AQ, is equal to the distance between P and Q minus the distance between R and P.

Distance between P and Q = 600 m
Distance between R and P = 300 m

Therefore, AQ = 600 m - 300 m = 300 m.

Now we can calculate the bearing:

Bearing = arctan(AD / AQ)
Bearing = arctan(600 m / 300 m)
Bearing = arctan(2)

Using a calculator, we find that arctan(2) is approximately 63.43 degrees.

So, the bearing of point P from point Q is approximately 63.43 degrees.

To find the distance between points R and Q, we can use the Pythagorean theorem:

Distance = sqrt((AQ^2) + (AD^2))
Distance = sqrt((300 m)^2 + (600 m)^2)
Distance = sqrt(90000 m^2 + 360000 m^2)
Distance = sqrt(450000 m^2)
Distance ≈ 670.82 m

Therefore, the distance between points R and Q is approximately 670.82 m.

Answer 3

To find the bearing of P from Q, we can use trigonometry. First, let's draw a diagram to visualize the problem:

P -------- Q
\
\
R

Given that R is 300m north of P, we can determine that the opposite side of the triangle created (PR) has a length of 300m. Additionally, we know that the adjacent side of the triangle (PQ) has a length of 600m.

We can find the bearing by using the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. Therefore, the tangent of the angle between P and Q is given by:

tan(angle) = PR / PQ = 300 / 600 = 1/2

To find the angle, we take the arctangent (inverse tangent) of both sides:

angle = arctan(1/2)

Using a calculator, we find that the angle is approximately 26.57 degrees. However, this angle is measured from the north direction.

Since we want to find the bearing of P from Q, we need to subtract this angle from 90 degrees (which points east). Therefore, the bearing of P from Q is:

90 - 26.57 ≈ 63.43 degrees.

To find the distance between R and Q, we can use the Pythagorean theorem. The distance (RQ) is equal to the square root of the sum of the squares of the two sides (PR and PQ):

RQ = √(PR² + PQ²)
= √(300² + 600²)
= √(90000 + 360000)
= √(450000)
= 670.82m

Therefore, the distance between R and Q is approximately 670.82 meters.