A traveller moves from town p on a bearing of 055 degree to a town q 200km away.he then moves from q on a bearing of 155 degree tn a town r 400km from q.find
A.distance between p and r
B.bearing of p from r,correct to the nearest degree
Solution
A traveller moves from town p on a HEADING of 055 degree to a town q 200km away.he then moves from q on a HEADING of 155 degree tn a town r 400km from q.find
A.distance between p and r
B.bearing of p from r,correct to the nearest degree
========================= ^&*% landlubber mathematicians
anyway
DRAW IT !!
In triangle PQR angle Q is 180 - (155-55) = 80
so
find length of side q opposite Q from P to R, law of cosines
q^2 = r^2 + p^2 - 2 p r cos 80 (almost a right triangle :)
q^2 = 200^2 + 400^2 - 2(200)(400)(0.174)
q^2 = 40,000 + 160,000 - 27,840 = 172,160
q = 415 km from p to r
now bearing
find angle in triangle at P with law of sines
sin 80/415 = sin P / 400
sin P = (400/415) sin 80 = 0.949
P = 71.1 deg
so bearing of R from P = 55+71.7 = 126.7
we want bearing of P from R so add 180 degrees
= 306.7 degrees so head 306.7 , about 37deg north of west to get home to P
it is very correct thanks
A. To find the distance between towns P and R, we can use the cosine rule:
c² = a² + b² - 2ab * cos(C)
Where:
- c is the distance between towns P and R
- a is the distance between towns P and Q (200km)
- b is the distance between towns Q and R (400km)
- C is the angle at town Q (180° - (155° - 55°) = 80°)
Using the formula:
c² = 200² + 400² - 2(200)(400) * cos(80°)
c² = 40,000 + 160,000 - 160,000 * cos(80°)
c² = 200,000 - 160,000 * cos(80°)
c² ≈ 200,000 - 160,000 * (-0.173648)
c² ≈ 200,000 + 27,709.76
c ≈ √227,709.76
c ≈ 477.56 km
Therefore, the distance between town P and town R is approximately 477.56 km.
B. To find the bearing of P from R, we can use the tangent rule.
tan(B) = (sin(A) * cos(C)) / (cos(A) * sin(C))
Where:
- B is the bearing from R to P
- A is the angle at town Q (155° - 55° = 100°)
- C is the angle at town R (180° + 155° - 180° = 155°)
Using the formula:
tan(B) = (sin(100°) * cos(155°)) / (cos(100°) * sin(155°))
tan(B) = (0.984808 * (-0.573576)) / (0.173648 * 0.987688)
tan(B) ≈ (-0.56410) / (0.17119)
tan(B) ≈ -3.2942
Taking the arctangent of both sides to find B:
B ≈ arctan(-3.2942)
B ≈ -72°
Therefore, the bearing of P from R, correct to the nearest degree, is approximately -72°.
To find the distance between towns P and R, we can use the cosine rule. The cosine rule states that in a 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 two sides multiplied by the cosine of the included angle.
Let's break down the information given:
1. The traveller moves from P on a bearing of 055 degrees to town Q, which is 200 km away.
2. The traveller then moves from Q on a bearing of 155 degrees to town R, which is 400 km away.
To solve for the distance between P and R:
Step 1: Calculate the distance between Q and R using the cosine rule.
Using the cosine rule: c² = a² + b² - 2ab * cos(C)
a = 200 km (distance from Q to P)
b = 400 km (distance from Q to R)
C = 155° (angle at Q)
c² = (200)² + (400)² - 2(200)(400) * cos(155°)
c² = 40,000 + 160,000 - 160,000 * cos(155°)
c² = 40,000 + 160,000 - (-115,657.55)
c² = 316,657.55
c ≈ √316,657.55
c ≈ 562.78 km
So, the distance between towns P and R is approximately 562.78 km.
To find the bearing of P from R:
Step 1: Calculate the absolute bearing from Q to R.
To find the absolute bearing, subtract the bearing angle from 180°.
Absolute Bearing = 180° - 155°
Absolute Bearing = 25°
Step 2: Calculate the relative bearing from P to R.
To calculate the relative bearing, subtract the absolute bearing from 360°.
Relative Bearing = 360° - 25°
Relative Bearing = 335°
Therefore, the bearing of P from R, correct to the nearest degree, is 335°.