An airplane flies from a town X on a bearing of N45
0E to another
town Y, a distance of 200km. It then. Changes course and flies to
another town Z on a bearing of S60
0E. If Z is directly east of X, calculate, correct to 3 significant figures. a. the distance from X to Z
b. the distance from Y to XZ
a. the distance XZ = 200cos45° + (200 sin45°)tan60° = 200/√2 + 200/√2 * √3
b. 200 cos45° = 200/√2
All angles are measured CW from +y-axis.
a. cos45 = h/200.
h = 141.4 km.
sin45 = d1/200.
d1 = 141.4 km.
Tan60 = d2/141.4.
d2 = 245 km.
XZ = d1 + d2 = 141.4 + 245 = 386 km.
b. YZ^2 = d2^2 + h^2 = 245^2 + 141.4^2 = 80,019,
YZ = 283 km[120o] CW.
b.
in a diagram AB = 8km BC= 13km the bearing of A from B = 310 and the bearing of B from C is 230 calculate the distance AC the bearing of A and C
To solve this problem, we can use basic trigonometry and the concept of bearings. Let's break down each part of the problem.
Given:
- An airplane flies from town X on a bearing of N45°E to town Y, a distance of 200 km.
- The airplane then changes course and flies to town Z on a bearing of S60°E.
- Z is directly east of X.
a. Calculating the distance from X to Z:
Since Z is directly east of X, we know that the angle between the line ZX and the north direction is 90°. Also, the bearing from X to Z is S60°E.
To find the distance from X to Z, we can use trigonometry. We can break down the triangle into two right-angled triangles: one with a vertical side and another with a horizontal side.
In the triangle with the vertical side, we have the angle between the line ZX and the north direction as 90°. The bearing from X to Z is S60°E, which means the angle between the line ZX and the east direction is 60°.
Using trigonometry, we can find the length of the vertical side (VZ) using the formula:
VZ = hypotenuse (ZX) * sin(angle)
Given:
- The distance from X to Z (ZX) is equal to 200 km.
Using the formula above, we can calculate VZ:
VZ = 200 km * sin(60°)
Now, let's calculate VZ:
VZ = 200 km * 0.866 (rounded to 3 decimal places)
VZ ≈ 173.2 km (rounded to 3 significant figures)
Therefore, the distance from X to Z is approximately 173.2 km (correct to 3 significant figures).
b. Calculating the distance from Y to XZ:
Since Y is the starting point, we need to find the length of the line XY first.
In the triangle with the vertical side, we can again use trigonometry to find the length of XY using the same formula:
XY = hypotenuse (YX) * sin(angle)
Given:
- The distance from X to Y (YX) is equal to 200 km.
- The angle between the line YX and the north direction is 45° (N45°E).
Using the formula above, we can calculate XY:
XY = 200 km * sin(45°)
Now, let's calculate XY:
XY = 200 km * 0.707 (rounded to 3 decimal places)
XY ≈ 141.4 km (rounded to 3 significant figures)
To find the distance from Y to XZ, we need to subtract the length of VZ (which we calculated earlier) from the length of XY:
YXZ = XY - VZ
YXZ = 141.4 km - 173.2 km (rounded to 3 significant figures)
YXZ ≈ -31.8 km (rounded to 3 significant figures)
However, it doesn't make sense to have a negative distance, so we can conclude that there might be an error in the given information or calculations.
Please double-check the problem and ensure the values are accurate.