An aeroplane flies due north from an airport after 800km it then travels on a bearing of 050degrees for 500km at which point it crises a River.calculate the distance and bearing of this point from the airport.
An aircraft flies due North from an airport. After 800 Kilometers, it then travels on a bearing of 050° for 500 Kilometers, at which point it crosses a village. Calculate the distance and bearing of the village from the airport?
Good
To calculate the distance and bearing of the point where the airplane crosses the river from the airport, we can use the concept of vector addition.
First, let's break down the airplane's journey into two separate legs:
1. The first leg is a due north flight from the airport for 800 km. This can be represented as a vector pointing directly north with a magnitude of 800 km.
2. The second leg is a flight on a bearing of 050 degrees (clockwise from north) for 500 km. To convert this bearing to a vector, we need to break it down into east (x-component) and north (y-component) components.
To calculate the x-component:
cos(050) = x / 500 km
x ≈ 500 km * cos(050) ≈ 500 km * 0.64278760968 ≈ 321.39 km
To calculate the y-component:
sin(050) = y / 500 km
y ≈ 500 km * sin(050) ≈ 500 km * 0.76604444311 ≈ 383.02 km
The vector for the second leg is approximately 321.39 km east and 383.02 km north.
Now, we can find the total displacement by adding the vectors of both legs.
To calculate the x-component:
Total x-component = 0 + 321.39 km ≈ 321.39 km
To calculate the y-component:
Total y-component = 800 km + 383.02 km ≈ 1183.02 km
The total displacement vector is approximately 321.39 km east and 1183.02 km north.
To find the distance between the airport and the point of intersection with the river, we can use the Pythagorean theorem:
Distance = sqrt((321.39 km)^2 + (1183.02 km)^2) ≈ sqrt(103289.2721 km^2 + 1397728.7204 km^2) ≈ sqrt(1501017.9925 km^2) ≈ 1225.96 km
Therefore, the distance between the airport and the point where the airplane crosses the river is approximately 1225.96 km.
To find the bearing of this point from the airport, we can use the inverse trigonometric functions:
Bearing = arctan((x-component / y-component)) ≈ arctan(321.39 km / 1183.02 km) ≈ arctan(0.2718) ≈ 15.68 degrees
Therefore, the bearing of the point where the airplane crosses the river from the airport is approximately 15.68 degrees.
To calculate the distance and bearing of the point where the airplane crosses the river from the airport, we can use vector addition and trigonometry.
1. Draw a diagram: Start by drawing a diagram representing the given information. Label the initial position of the airplane as the airport, and mark the distance of 800km traveled due north. Then, draw a line from the airport in the direction of the bearing 050 degrees and mark the distance of 500km.
2. Break down the vectors: Break down the two vectors (north and bearing 050) into their horizontal (West-East) and vertical (South-North) components.
- For the north vector, the vertical component (north) is 800km and the horizontal component (west-east) is 0km.
- For the bearing 050 vector, the vertical component (north) can be calculated using trigonometry. Since the bearing is measured clockwise from the north direction, the vertical component is given by sin(90 - bearing) x distance. In this case, sin(90 - 50) x 500km.
Calculate the vertical component:
sin(90 - 50) x 500km = sin(40) x 500km = 0.642 x 500km ≈ 321.1km (rounded to one decimal place)
The horizontal component for the bearing 050 vector is given by cos(90 - bearing) x distance. In this case, cos(90 - 50) x 500km.
Calculate the horizontal component:
cos(90 - 50) x 500km = cos(40) x 500km = 0.766 x 500km ≈ 383.0km (rounded to one decimal place)
3. Add the components: Add the horizontal and vertical components of the vectors to get the resulting displacement from the airport to where the plane crosses the river.
Horizontal component: 0km (from plane flying north) + 383.0km (from plane flying on bearing 050) = 383.0km
Vertical component: 800km (from plane flying north) + 321.1km (from plane flying on bearing 050) = 1121.1km
Therefore, the resulting displacement is approximately 383.0km horizontally and 1121.1km vertically.
4. Calculate the distance: Use the Pythagorean theorem to calculate the straight-line distance between the airport and the point where the plane crosses the river.
Distance = √(horizontal^2 + vertical^2)
Distance = √(383.0km^2 + 1121.1km^2)
Note: Make sure to use the exact values obtained from the calculations rather than rounded values.
5. Calculate the bearing: Use trigonometry to calculate the bearing angle between the airport and the point where the plane crosses the river.
Bearing = tan^-1(vertical / horizontal)
Bearing = tan^-1(1121.1km / 383.0km)
Note: This will give the value in radians. To convert it to degrees, multiply by 180/π.
So, to summarize:
- The distance between the airport and the point where the plane crosses the river is given by the calculated distance.
- The bearing of this point from the airport is given by the calculated bearing.