an aeroplane leaves an airport A an flies on a bearing of N41E for 1 and half hour at 699km/h to airport B.it then flies on a bearing of S50E for 3 and half hours at 400km/h to an airport C.calculate the distance from C to A
planes fly on headings, not bearings.
Use the law of cosines:
CA^2 = AB^2 + BC^2 - 2*AB*BC*cos(41+50)°
You can easily get AB and BC using distance = speed * time.
To calculate the distance from airport C to airport A, we can use the concept of vectors and vector addition.
First, let's find the displacement from airport A to airport B. The plane flies on a bearing of N41E for 1.5 hours at 699 km/h. We can calculate the displacement using the formula:
Displacement = Speed x Time
Displacement from A to B = 699 km/h x 1.5 hours = 1,048.5 km
Next, let's find the displacement from airport B to airport C. The plane flies on a bearing of S50E for 3.5 hours at 400 km/h. Again, using the same formula:
Displacement from B to C = 400 km/h x 3.5 hours = 1,400 km
Now, we can add the two displacements together to find the total displacement from airport C to airport A. Since the displacements are on different bearings, we need to break them down into their north-south (N/S) and east-west (E/W) components separately.
For the displacement from A to B:
- The N/S component is given by the north component -1,048.5 km * sin(41°) ≈ -694.63 km
- The E/W component is given by the east component 1,048.5 km * cos(41°) ≈ 807.72 km
For the displacement from B to C:
- The N/S component is given by the south component 1,400 km * sin(50°) ≈ 1,072.37 km
- The E/W component is given by the east component 1,400 km * cos(50°) ≈ 1,074.61 km
To find the total displacement:
- The N/S component is the sum of the N/S components of the two displacements: -694.63 km + 1,072.37 km ≈ 377.74 km (North)
- The E/W component is the sum of the E/W components of the two displacements: 807.72 km + 1,074.61 km ≈ 1,882.33 km (East)
Now, we can use the Pythagorean theorem to find the magnitude of the total displacement, which will give us the distance from airport C to airport A:
Distance from C to A = √(N/S component^2 + E/W component^2)
= √(377.74 km^2 + 1,882.33 km^2)
≈ √(142,483.28 km^2)
≈ 377.5 km
Therefore, the distance from airport C to airport A is approximately 377.5 kilometers.