To answer the first question, you can start by drawing a diagram. Let's use a coordinate system with Toronto at the origin (0,0). Pittsburgh is 350 km due south, so its coordinates are (0, -350). Philadelphia is 390 km due west, so its coordinates are (-390, 0).
Next, draw the vector representing the wind. Since the wind is blowing from the east at 60 km/h, the vector representing the wind will point west with a magnitude of 60 km/h.
To determine the direction that the pilot of Plane A must point the plane, you need to find the resultant vector of the plane's airspeed and the wind velocity. You can do this by vector addition.
The airspeed of Plane A is 400 km/h, and the wind velocity is 60 km/h to the west. Since the wind is blowing against the plane, you subtract the wind velocity vector from the airspeed vector.
To compute the resultant vector, subtract the x-components and y-components separately:
x-component: 400 km/h - 60 km/h = 340 km/h (to the east)
y-component: 0 km/h (there is no wind blowing north or south)
Hence, the pilot of Plane A must point the plane in the east direction.
For the second question, to find the time taken for the entire flight for Plane A, you have to calculate the distance between Toronto and Pittsburgh and add it to the distance between Pittsburgh and Philadelphia.
The distance between Toronto and Pittsburgh is the y-component of the position vector for Pittsburgh, which is 350 km.
The distance between Pittsburgh and Philadelphia is the x-component of the position vector for Philadelphia, which is 390 km.
The total distance traveled is: 350 km + 390 km = 740 km.
To find the time taken, you can divide the total distance by the airspeed of Plane A: 740 km ÷ 400 km/h = 1.85 hours.
Since there is a 0.50-hour layover in Pittsburgh, the entire flight for Plane A will take 1.85 hours + 0.50 hours = 2.35 hours.
For the third question, the time that the pilot of Plane B must wait before leaving Toronto can be determined by calculating the time taken for Plane A to complete its journey.
From the previous calculation, we found that the total flight time for Plane A is 2.35 hours.
To arrive in Philadelphia at the same time as Plane A, Plane B needs to depart Toronto 2.35 hours minus the time taken to travel from Toronto to Pittsburgh.
The time taken to travel from Toronto to Pittsburgh can be found by dividing the distance by the airspeed: 350 km ÷ 400 km/h = 0.875 hours.
Therefore, Plane B needs to wait 2.35 hours - 0.875 hours = 1.475 hours (or approximately 1 hour and 28 minutes) before leaving Toronto in order to arrive in Philadelphia at the same time as Plane A.