A small plane is 150 km south of the equator. The plane is flying at 150 km/h at a heading of 30∘ to the west of north. how many minutes will the plane cross the equator?
We have a right triangle ABC, CA lies on the equator, where C is the destination (where plane intercepts the equator).
B is the initial position of the plane, and is immediately south of A.
So AB=150 km, ∠ABC=30°.
We need length of BC.
Using definition of cosine,
AB/BC=cos(30)=AB/BC
BC=AB/cos(30)=150/(sqrt(3)/2)=173.21 km
Time required = distance/speed
=150/(sqrt(3)/2) km / 150 km/hr
=2/sqrt(3) hours
=1.155 hours.
Note:
Real life situation is that all distances are on the surface of a sphere of radius 6471 km. The actual calculations used by navigators require spherical trigonometry. In this case, the "spherical triangle" is relatively small compared to the radius of the earth, and the difference in distance amounts to only about 0.01 km.
To determine how many minutes it will take for the plane to cross the equator, we need to calculate the distance it needs to cover.
First, let's find the horizontal distance traveled by the plane. We can use trigonometry to do this.
The vertical component of the velocity vector can be calculated by taking the sine of the angle: V_vertical = V * sin(angle).
V_vertical = 150 km/h * sin(30°)
V_vertical = 150 km/h * 0.5
V_vertical = 75 km/h
Since the plane is flying south of the equator, the vertical velocity component should be negative. Therefore, the horizontal velocity component will be the opposite.
V_horizontal = -75 km/h
The time it takes for the plane to cross the equator can be calculated by dividing the distance traveled by the horizontal velocity.
Distance = 150 km (since the plane is 150 km south of the equator)
Velocity = -75 km/h
Time = Distance / Velocity
Time = 150 km / -75 km/h
Time = -2 h
Note that the negative sign indicates that the plane is traveling in the opposite direction. To convert hours to minutes, we can multiply the result by 60 since there are 60 minutes in an hour.
Time in minutes = (-2 h) * 60 min/h
Time in minutes = -120 min
Therefore, the plane will cross the equator in 120 minutes, or 2 hours, in the opposite direction.