An aeroplane needs to fly due east from city A to city B at a speed of 400km/hr. However a 50km/hr wind blows constantly from the north. In what direction must aeroplane head and at what speed must it travel?

(1) draw vectors for speed direction

adding vectors head-to-tail
Results: a right angle
(2) draw the resultant vector (hypothenuse) completing the right triangle.
(30 Solve using pythag. theorem

To find the direction and speed at which the airplane must travel, we need to consider the effect of the wind on the airplane's motion.

Let's first visualize the situation. We have the following information:
- The airplane needs to fly due east from city A to city B.
- The airplane has a speed of 400 km/hr relative to the air (we'll call this its airspeed).
- There is a constant wind blowing from the north at a speed of 50 km/hr.

To find the direction the airplane must head, we need to consider the wind's effect on the airplane's motion. The wind will create a force on the airplane, pushing it off course.

Since the wind is blowing from the north, its force on the airplane will oppose the airplane's eastward motion. Therefore, the airplane needs to head slightly to the north of east to counteract the wind and maintain a due east course.

To calculate the angle at which the airplane must head, we can use basic trigonometry. Let's call this angle θ.

In a right-angled triangle, the sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. In this case, the opposite side is the northward component of the wind speed (50 km/hr) and the hypotenuse is the airplane's airspeed (400 km/hr).

So, sin(θ) = opposite/hypotenuse = 50/400 = 1/8.

Taking the inverse sine (or arcsine) of both sides, θ = arcsin(1/8).

Using a calculator, we find that θ ≈ 7.1 degrees.

Therefore, the airplane must head approximately 7.1 degrees north of east to counteract the wind and maintain a due east course.

Now, let's calculate the speed at which the airplane must travel. Since the airplane's airspeed is 400 km/hr and it needs to counteract a 50 km/hr wind, the resulting ground speed (the speed of the airplane relative to the ground) will be the vector sum of these two velocities.

Using the Pythagorean theorem, ground speed = √(airspeed^2 + wind speed^2).

Substituting the values, ground speed = √(400^2 + 50^2) ≈ √(160000 + 2500) ≈ √162500.

Calculating the square root, ground speed ≈ 403.11 km/hr.

Therefore, the airplane must travel at a ground speed of approximately 403.11 km/hr to maintain a due east course in the presence of the 50 km/hr wind blowing from the north.