A plane starts from point P and flies 240km on a bearing of 120 degrees to Q. At Q the plane makes a 90 degree turn and flies 100km on a bearing of 210 degrees to R. At R the plane again changes directions and flies back to P.
Is there a question somewhere in that?
To find the distance and bearing of the plane's final destination (point P), we can break down the given information into steps:
Step 1: Determine the coordinates of each point (P, Q, R) using distance and bearing.
Step 2: Calculate the distance between each pair of consecutive points (PQ, QR, RP).
Step 3: Find the bearing (direction) from each point to the next (PQ, QR, RP).
Step 4: Determine the total distance traveled by summing the distances (PQ + QR + RP).
Step 5: Calculate the overall bearing from the final point back to the starting point (P).
Let's go through each step in detail:
Step 1: Determine the coordinates of each point (P, Q, R) using distance and bearing.
Given:
- Plane flies 240km on a bearing of 120 degrees from P to Q.
- Plane then flies 100km on a bearing of 210 degrees from Q to R.
We can assume that point P is the reference point with coordinates (0, 0).
To find the coordinates of Q, we can use the distance (240km) and bearing (120 degrees) from P to Q.
Using trigonometry, we can calculate the vertical and horizontal components:
Vertical component (y-coordinate of Q) = 240km * sin(120 degrees)
Horizontal component (x-coordinate of Q) = 240km * cos(120 degrees)
To find the coordinates of R, we'll use the distance (100km) and bearing (210 degrees) from Q to R. Since Q is not the reference point, we need to add the coordinates of Q to the components of R.
Vertical component (y-coordinate of R) = y-coordinate of Q + 100km * sin(210 degrees)
Horizontal component (x-coordinate of R) = x-coordinate of Q + 100km * cos(210 degrees)
Step 2: Calculate the distance between each pair of consecutive points (PQ, QR, RP).
Distance PQ = 240km (given)
Distance QR = 100km (given)
Distance RP = distance from R to P, which can be calculated using the coordinates of both points.
Distance RP = sqrt((x-coordinate of R - x-coordinate of P)^2 + (y-coordinate of R - y-coordinate of P)^2)
Step 3: Find the bearing (direction) from each point to the next (PQ, QR, RP).
Bearing PQ = 120 degrees (given)
Bearing QR = 90 degrees (given) since the plane makes a 90-degree turn at Q.
Bearing RP = bearing from R to P, which can be calculated using the coordinates of both points.
Bearing RP = atan2(y-coordinate of P - y-coordinate of R, x-coordinate of P - x-coordinate of R)
Step 4: Determine the total distance traveled by summing the distances (PQ + QR + RP).
Total distance traveled = Distance PQ + Distance QR + Distance RP
Step 5: Calculate the overall bearing from the final point back to the starting point (P).
Overall bearing from R to P can be calculated using the coordinates of both points:
Overall bearing from R to P = atan2(y-coordinate of P - y-coordinate of R, x-coordinate of P - x-coordinate of R)
Finally, once you have all the values from the calculations, you can provide the distance and bearing of the plane's final destination (point P) as the output of the problem.