A plane flew 150 miles on a course of 220 degrees and then 130 miles on a course of 130 degrees. Then the plane returned to it's starting point via the shortest route possible. Find that shortest distance.
The individual displacements are
150 @ 220° = (-96.4,-114.9)
130 @ 130° = (99.6,-83.6)
So, the final location is at (3.2,-198.5)
Just find the distance from there to the origin.
To find the shortest distance, we can break down the problem into smaller steps:
Step 1: Determine the coordinates of the starting point, the first destination, and the second destination.
- Let's assume the starting point is at coordinates (0, 0).
- The first destination is 150 miles away on a course of 220 degrees.
- The second destination is 130 miles away on a course of 130 degrees.
Step 2: Convert the distances and angles into Cartesian coordinates.
- Use trigonometry to find the change in x and y coordinates.
- For the first destination:
- Change in x = 150 * cos(220 degrees)
- Change in y = 150 * sin(220 degrees)
- For the second destination:
- Change in x = 130 * cos(130 degrees)
- Change in y = 130 * sin(130 degrees)
Step 3: Calculate the final coordinates after reaching both destinations.
- Add the changes in x and y coordinates to the starting point's coordinates.
- For the first destination:
- Final coordinates after reaching the first destination: (0 + change in x, 0 + change in y)
- For the second destination:
- Final coordinates after reaching the second destination: (0 + change in x + change in x, 0 + change in y + change in y)
Step 4: Calculate the distance between the final coordinates and the starting point.
- Find the Euclidean distance using the Pythagorean theorem.
- Distance = sqrt((final x - starting x)^2 + (final y - starting y)^2)
Let's perform these steps to find the shortest distance.
To find the shortest distance back to the starting point, we can use the concept of vectors. Let's break down the steps to solve this problem:
1. Convert the given distances and courses into vectors:
- The first leg of the flight can be represented as a vector with a magnitude of 150 miles and a direction of 220 degrees.
- The second leg of the flight can be represented as a vector with a magnitude of 130 miles and a direction of 130 degrees.
2. Find the initial and final positions of the plane:
- The initial position of the plane can be considered as the origin (0, 0) on a Cartesian coordinate system.
- The final position of the plane can be found by adding both vectors to the initial position.
3. Find the displacement vector from the initial to the final position:
- Since the plane returns to its starting point, the displacement vector will be the negative of the final position vector.
4. Calculate the magnitude of the displacement vector:
- The magnitude of a vector can be calculated using the formula: magnitude = √(x^2 + y^2), where x and y are the Cartesian coordinates of the vector.
5. Find the shortest distance using the magnitude of the displacement vector:
- The shortest distance is equal to the magnitude of the displacement vector.
Let's now perform the calculations step by step:
1. Convert the given distances and courses into vectors:
- The first vector will be represented as V1 = 150 * cos(220°)i + 150 * sin(220°)j.
- The second vector will be represented as V2 = 130 * cos(130°)i + 130 * sin(130°)j.
2. Find the initial and final positions of the plane:
- The initial position of the plane is (0, 0).
- The final position of the plane can be found by adding both vectors:
Final position = (V1 + V2)
3. Find the displacement vector from the initial to the final position:
- The displacement vector is the negative of the final position vector:
Displacement vector = -(Final position)
4. Calculate the magnitude of the displacement vector:
- The magnitude of the displacement vector is given by:
Magnitude = √(x^2 + y^2), where x and y are the Cartesian coordinates of the displacement vector.
5. Find the shortest distance using the magnitude of the displacement vector:
- The shortest distance is equal to the magnitude of the displacement vector.
By following these steps and performing the necessary calculations, you will find the shortest distance the plane must travel to return to its starting point.