1. A boy moves from a point X and walks 285m to Y on a bearing of 078° . He then moves due south to a point Z, which is 307m from X. Calculate:
a) the bearing of X from Z
b) the distance between YZ
Using the law of cosines,
307^2 = 285^2 + YZ^2 - 2*285*YZ*cos78°
YZ = 187.84
In triangle XYZ, find angle Z using the law of sines:
sinZ/285 = sin78°/307
Z = 65°
So, the bearing of X from Z is 360-65 = 295°
C0rrect
Correct
Nice one
Diagram
a) The bearing of X from Z can be calculated by finding the bearing of Z from X and then adding 180° to it, since it is in the opposite direction.
To find the bearing of Z from X, we need to find the angle between the line ZX and the north direction. Since ZX is due south, the angle is 90°.
Adding 180° to the bearing of Z from X gives us the bearing of X from Z, which is 90° + 180° = 270°.
Therefore, the bearing of X from Z is 270°.
b) To find the distance between Y and Z, we can use the Pythagorean theorem since Y and Z form a right triangle with X as the right angle.
The distance between Y and Z is the hypotenuse of this triangle.
Using the formula c^2 = a^2 + b^2, where c is the hypotenuse and a and b are the other two sides, we have:
c^2 = 285^2 + 307^2
c^2 = 81225 + 94249
c^2 = 175474
c = √175474
c ≈ 419.08
Therefore, the distance between Y and Z is approximately 419.08 meters.
To find the answers to these questions, we can use trigonometry and the concept of bearings.
a) Bearing of X from Z:
To find the bearing of X from Z, we need to determine the angle between the line XZ and the north direction.
First, we find the angle between the line XY and the north direction. The bearing of Y from X is given as 078°. Since bearings are measured clockwise from the north direction, the angle between XY and the north direction is 360° - 078° = 282°.
Next, we know that YZ is due south, which means it is in a straight line vertically downward. The angle between YZ and the north direction is 180°.
To find the bearing of X from Z, we need to find the angle between XZ and the north direction. This can be done by subtracting the angle between XY and the north direction (282°) from the angle between YZ and the north direction (180°).
Bearing of X from Z = 180° - 282° = -102°
Note: In most cases, bearings are measured as positive angles from the north direction. However, while calculating bearings, we sometimes encounter negative values. In this case, the negative value (-102°) represents a bearing with respect to the south direction.
b) Distance between Y and Z:
To find the distance between Y and Z, we can use the Pythagorean theorem.
We know that YZ is the hypotenuse of a right-angled triangle with two perpendicular sides XY and XZ. We have the length of XY as 285m and XZ as 307m.
Using the Pythagorean theorem:
YZ^2 = XY^2 + XZ^2
YZ^2 = 285^2 + 307^2
YZ^2 = 81225 + 94249
YZ^2 = 175474
Taking the square root of both sides:
YZ = √175474
YZ ≈ 418.99m
Therefore, the distance between Y and Z is approximately 418.99 meters.