Multiple choice: An airplane flying due east at 539 mph in still air, encounters a 61-mph tail wind acting in the direction 45 degrees north of east. The airplane holds its compass heading but because of the wind acquires a new ground speed and direction. What is its new direction?

a) 5 degrees
b) 85.8 degrees
c) 4.2 degrees
d) 85 degrees

add the two vectors and you get

<539,0> + <43,43> = <582,43>
arctan(43/582) = 4.2°

As a bearing, that's E 4.2° N, or 85.8°

To find the new direction of the airplane, we can use vector addition.

First, let's break down the velocities into their x and y components.

The original velocity of the airplane flying due east at 539 mph can be written as (539 mph, 0 mph). Since it is flying due east, its y-component is zero.

The tailwind acts in the direction 45 degrees north of east with a speed of 61 mph. To find its x and y components, we can use trigonometry.

The x-component of the tailwind velocity is given by the cosine of the angle: 61 mph * cos(45 degrees) = 43.1 mph.
The y-component of the tailwind velocity is given by the sine of the angle: 61 mph * sin(45 degrees) = 43.1 mph.

Now we can add the components of the original velocity and the tailwind velocity together to find the new velocity.

Adding the x-components: 539 mph + 43.1 mph = 582.1 mph.
Adding the y-components: 0 mph + 43.1 mph = 43.1 mph.

Therefore, the new velocity of the airplane is (582.1 mph, 43.1 mph).

To find the direction of this vector, we can use trigonometry again. The direction can be found using the arctan function, which gives us the angle in degrees.

The angle can be calculated as arctan(43.1 mph / 582.1 mph) ≈ 4.2 degrees.

Hence, the new direction of the airplane is approximately 4.2 degrees.

Therefore, the correct answer is c) 4.2 degrees.