Question

An aircraft flew from port A on a bearing of 130° to another port B 50km apart. It

then flew from port B on a bearing of 040° to port C, a distance of120km. calcula
i)
ii)
The distance of port A from port C to the nearest kilometer.
the bearing of port A from port C to the nearest distance.

Answer 1

Answer: 248°

Explanation:

Let's solve this step by step:

i) To find the distance of port A from port C, you can use the law of cosines since you have two sides and the included angle. We'll call the distance AC "d."

d² = 50² + 120² - 2(50)(120)cos(130° - 40°)

First, find the cosine of the difference between the two bearings:

cos(130° - 40°) = cos(90°) = 0

Now, plug this into the equation:

d² = 50² + 120² - 2(50)(120)(0)

d² = 2500 + 14400 - 0

d² = 16900

Now, take the square root:

d = √16900 = 130 km

So, the distance of port A from port C is 130 km to the nearest kilometer.

ii) To find the bearing of port A from port C, you can use trigonometry. Since you've already found the distance AC (130 km), you can use the law of sines to find an angle. Let's call this angle θ.

sin(θ)/120 = sin(130°)/130

Now, solve for sin(θ):

sin(θ) = (120/130) * sin(130°)

sin(θ) ≈ 0.9257

Now, find θ:

θ ≈ arcsin(0.9257) ≈ 67.99°

However, since you want the bearing from port C to port A, you need to add 180° to this angle:

θ (bearing) ≈ 67.99° + 180° ≈ 247.99°

So, the bearing of port A from port C to the nearest degree is approximately 248°.