Question

A plane flies 200km on a bearing 032° it then flies 350km on a bearing 275°
Find the bearing and distance of the plane from it's starting point

Answer 1

To find the distance and bearing of the plane from its starting point, we will use the Law of Sines and trigonometry.

First, we can break down the two displacements of the plane into their horizontal and vertical components:

For the first leg:
Horizontal component: 200km * cos(032°) = 200km * 0.8387 = 167.74km
Vertical component: 200km * sin(032°) = 200km * 0.5446 = 108.92km

For the second leg:
Horizontal component: 350km * cos(275°) = 350km * -0.9659 = -337.06km
Vertical component: 350km * sin(275°) = 350km * -0.2588 = -90.34km

Next, we can find the total horizontal and vertical displacements of the plane:
Total horizontal displacement = 167.74km - 337.06km = -169.32km
Total vertical displacement = 108.92km - 90.34km = 18.58km

Now, we can use trigonometry to find the distance of the plane from its starting point:
Distance = √((-169.32)^2 + 18.58^2) ≈ 171.3km

Lastly, we can find the bearing of the plane from its starting point:
Bearing = arctan(18.58 / -169.32) ≈ -6.26°

Therefore, the plane is approximately 171.3km away from its starting point on a bearing of 006°.

Answer 2

since bearings are measured clockwise from vertical, you have, for a distance d on a heading of θ,
horizontal component = d sinθ
vertical component = d cosθ

Answer 3

My apologies for the mistake in the previous calculation. Let's correct it using the correct horizontal and vertical components as per your clarification.

For the first leg:
Horizontal component: 200km * sin(032°) = 200km * 0.5299 = 105.98km
Vertical component: 200km * cos(032°) = 200km * 0.8479 = 169.58km

For the second leg:
Horizontal component: 350km * sin(275°) = 350km * -0.7832 = -274.62km
Vertical component: 350km * cos(275°) = 350km * 0.6216 = 217.56km

Now, we can find the total horizontal and vertical displacements of the plane:
Total horizontal displacement = 105.98km - 274.62km = -168.64km
Total vertical displacement = 169.58km + 217.56km = 387.14km

Now, we can use trigonometry to find the distance of the plane from its starting point:
Distance = √((-168.64)^2 + 387.14^2) ≈ 423.67km

Lastly, we can find the bearing of the plane from its starting point:
Bearing = arctan(168.64 / 387.14) ≈ 22.01°

Therefore, the plane is approximately 423.67km away from its starting point on a bearing of 022°.