A boy moves from point X and walks 285m to Y on a bearing of 078° . then moves due south to a point Z which is 307m from X . calculate:
(a) the bearing of X from Y
(b) the distance between Y,Z
draw triangle XYZ. Then using the law of sines,
sinZ/285 = sin78°/307
The bearing of X from Y is 360-Z
Now you know angles Y and Z, so X is easy.
use the law of cosines to find YZ:
YZ^2 = 285^2 + 307^2 - 2*285*307 cosX
To solve this problem, we will use the concepts of trigonometry and bearings.
(a) To find the bearing of X from Y, we need to calculate the angle between the line XY and the north direction.
1. Calculate the difference in the eastward (E) direction between points X and Y:
E = 285m * sin(078°)
2. Calculate the difference in the northward (N) direction between points X and Y:
N = 285m * cos(078°)
3. Calculate the bearing of X from Y:
Bearing_XY = arctan(E / N)
(b) To find the distance between points Y and Z, we will use Pythagoras' theorem.
1. Calculate the distance between points Y and Z:
Distance_YZ = sqrt((285m)^2 + (307m)^2)
Now, let's calculate the answers step by step.
(a) Calculate the bearing of X from Y:
1. Calculate the difference in the eastward (E) direction between points X and Y:
E = 285m * sin(078°)
E ≈ 285m * 0.97815
E ≈ 278.03625m
2. Calculate the difference in the northward (N) direction between points X and Y:
N = 285m * cos(078°)
N ≈ 285m * 0.20791
N ≈ 59.09835m
3. Calculate the bearing of X from Y:
Bearing_XY = arctan(E / N)
Bearing_XY ≈ arctan(278.03625m / 59.09835m)
Bearing_XY ≈ 78.873°
Therefore, the bearing of X from Y is approximately 78.873°.
(b) Calculate the distance between points Y and Z:
1. Calculate the distance between points Y and Z:
Distance_YZ = sqrt((285m)^2 + (307m)^2)
Distance_YZ = sqrt(81225m^2 + 94249m^2)
Distance_YZ = sqrt(6590832225 + 8880561441)
Distance_YZ = sqrt(15411343666)
Distance_YZ ≈ 124113.828m
Therefore, the distance between points Y and Z is approximately 124113.828m.
To solve this problem, we can use basic trigonometry and compass bearings.
(a) The bearing of X from Y:
To find the bearing of X from Y, we can subtract the given bearing of 078° from 180° since it is in the opposite direction (180° represents a south bearing).
Therefore, the bearing of X from Y is 180° - 078° = 102°.
(b) The distance between Y and Z:
To find the distance between Y and Z, we can use the Pythagorean theorem. Since the boy moves due south from Y to Z, we can consider this as the vertical component, and the distance from X to Y as the horizontal component of a right-angled triangle.
Using the Pythagorean theorem:
Distance^2 = (285m)^2 + (307m)^2
Distance^2 = 81225m^2 + 94249m^2
Distance^2 = 175474m^2
To find the distance, take the square root of both sides:
Distance = sqrt(175474m^2)
Distance ≈ 418.71m
Therefore, the distance between Y and Z is approximately 418.71m.