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

(a)distance between q and r
(b)bearing of r from q

Ok, the angle at p is between 310 and 075,or an angle of 125 (check that). YOu have two sides, one angle

find another angle with the law of sines.
then find the last angle (sum of angles is 180)
then find the last side by law of sines.

Make certain you draw the figure, life is simpler with a drawing.

thank you, although am still hooked

254

All angles are measured CW from +y-axis

a. PQ + QR = PR.
QR = PR - PQ = 90[310o] - 50[75o],
QR = (90*sin310-50*sin75) + (90*cos310-50*cos75)i,
QR = -117.2 + 44.9i = 125.5km[-69o] = 125.5km[291o] CW.

b. Bearing = 291o CW.

To find the distance between towns q and r, we can use the Law of Cosines. The Law of Cosines states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the lengths of those sides multiplied by the cosine of the included angle.

(a) Distance between q and r:
Let's label the side connecting q and r as side "x". We can use the Law of Cosines to find the value of x.

The formula for the Law of Cosines is:
c^2 = a^2 + b^2 - 2ab * cos(C)

In this case, a = 50 km (distance between p and q), b = 90 km (distance between p and r), and the angle C is the angle between b and c.

We can calculate the angle C using the given bearings:
Angle C = 310° - 75° = 235°

Now we can substitute these values into the Law of Cosines formula:
x^2 = 50^2 + 90^2 - 2 * 50 * 90 * cos(235°)

To calculate the distance, we need to evaluate this equation:
x^2 = 2500 + 8100 - 9000 * (-0.45399)
x^2 = 2500 + 8100 + 4085.91
x^2 ≈ 14,685.91

Taking the square root of both sides, we find:
x ≈ √14,685.91
x ≈ 121.16 km

Therefore, the distance between q and r is approximately 121.16 km.

(b) Bearing of r from q:
To calculate the bearing of r from q, we can use the concept of the bearing between two points. The bearing represents the angle formed between the line connecting two points and the north direction.

To find the bearing of r from q, we need to determine the direction from q to r using the concept of bearing angles.

First, we calculate the angle formed by the line connecting q and r with respect to the north direction:
Angle N to R = Bearing of q from p + Bearing of r from p

Angle N to R = 075 + 310
Angle N to R = 385°

Since the bearing is measured clockwise from the north direction, we subtract Angle N to R from 360° to find the bearing of r from q:
Bearing of r from q = 360° - Angle N to R

Bearing of r from q = 360° - 385°
Bearing of r from q ≈ -25°

Therefore, the bearing of r from q is approximately -25°.