An aircraft flies 500km from an airport A on a bearing of 070 degree to arrive at airport B.The plane then flies 800km from airport B on a bearing of 035 degree to arrive at airport C. Calculate the direct distance from A to C?
To find the direct distance from A to C, we can break down the journey into two right-angled triangles formed by the straight-line paths from A to B and from B to C.
Let's denote:
- Distance from A to B as AB = 500km
- Distance from B to C as BC = 800km
- Bearing from A to B as 070 degrees
- Bearing from B to C as 035 degrees
First, we'll find the horizontal and vertical components of AB:
Horizontal component of AB = AB * cos(180 - 70) = 500 * cos(110 degrees) ≈ 500 * (-0.3420) ≈ -171.0km
Vertical component of AB = AB * sin(180 - 70) = 500 * sin(110 degrees) ≈ 500 * 0.9397 ≈ 469.9km
Now, let's find the horizontal and vertical components of BC:
Horizontal component of BC = BC * cos(180 - 35) = 800 * cos(145 degrees) ≈ 800 * (-0.8192) ≈ -655.3km
Vertical component of BC = BC * sin(180 - 35) = 800 * sin(145 degrees) ≈ 800 * 0.5736 ≈ 458.9km
Since the horizontal components are negative, we need to subtract their absolute values:
Horizontal distance = |horizontal component of AB| + |horizontal component of BC| = 171.0 + 655.3 = 826.3km
Vertical distance = vertical component of AB + vertical component of BC = 469.9 + 458.9 = 928.8km
Now, we can find the direct distance from A to C using the Pythagorean theorem:
Distance from A to C = sqrt((horizontal distance)^2 + (vertical distance)^2)
Distance from A to C = sqrt((826.3)^2 + (928.8)^2)
Distance from A to C = sqrt(682080.2 + 862681.4)
Distance from A to C ≈ sqrt(1544761.6)
Distance from A to C ≈ 1242.6km
Therefore, the direct distance from airport A to airport C is approximately 1242.6km.