An airplane, flying at a speed of 420 miles per hour, flies from point A in the direction 131 degrees for 90 min. and then flies in the direction 41 degrees for 30 min. in what direction does the plane need to fly in order to get back to point A?
the speed is irrelevant, since it does not change. So the relative times are the same as the relative distances.
90 min at 131° moves the plane 90(cos41°,-sin41°) = (67.92,-59.04)
30 min at 41° moves it an additional 30(cos41°,sin41°) = (22.64,19.68)
So, total displacement is (90.56,-39.36)
tanθ = 90.56/39.36 = 2.30
θ = 66.5°
the heading back to A is thus 360-66.5 = 293.5°
To find the direction the plane needs to fly in order to get back to point A, we can start by finding the total distance the plane has traveled in each direction.
The first leg of the flight is for 90 minutes at a speed of 420 miles per hour. To find the distance traveled in this leg, we can use the formula:
Distance = Speed × Time
Distance = 420 miles/hour × 90 minutes × (1 hour / 60 minutes) = 630 miles
Similarly, the second leg of the flight is for 30 minutes at a speed of 420 miles per hour. Using the same formula, the distance traveled in this leg is:
Distance = 420 miles/hour × 30 minutes × (1 hour / 60 minutes) = 210 miles
Now, let's find the total displacement of the plane. Displacement is the straight-line distance and direction from the starting point (point A) to the ending point.
To find the total displacement, we can break down the distances traveled in each leg into their respective x and y components. We can then sum up the x and y components separately to find the total displacement in each direction.
First, let's break down the first leg.
Displacement in x-direction = Distance × cos(angle)
Displacement in x-direction = 630 miles × cos(131 degrees)
Displacement in y-direction = Distance × sin(angle)
Displacement in y-direction = 630 miles × sin(131 degrees)
Next, let's break down the second leg.
Displacement in x-direction = Distance × cos(angle)
Displacement in x-direction = 210 miles × cos(41 degrees)
Displacement in y-direction = Distance × sin(angle)
Displacement in y-direction = 210 miles × sin(41 degrees)
Now, let's add up the x and y components to find the total displacement in each direction.
Total displacement in x-direction = (630 miles × cos(131 degrees)) + (210 miles × cos(41 degrees))
Total displacement in y-direction = (630 miles × sin(131 degrees)) + (210 miles × sin(41 degrees))
Finally, we can find the angle of the total displacement using the formula:
Angle = atan2(total displacement in y-direction, total displacement in x-direction)
This will give us the direction the plane needs to fly in order to get back to point A.