a plane needs to fly at 40 degrees n of e when flying through a wind that blows 80 mph @ 70 degrees S of E. If the plane goes 320 mph in still air in what direction should the plane aim?

To determine the direction in which the plane should aim, we need to calculate the resulting ground speed and direction when considering the speed and direction of the wind.

Let's break down the given information:

1. The plane wants to fly at 40 degrees N of E.
2. The wind is blowing at 80 mph and is 70 degrees S of E.
3. The plane's speed in still air is 320 mph.

First, we need to find the components of the wind vector. We can use trigonometry to do this.

The eastward component of the wind can be calculated by multiplying the wind speed (80 mph) by the cosine of the angle between the wind direction and the east (E) direction. In this case, the angle is 70 degrees.

Eastward component of wind = 80 mph * cos(70 degrees)

Next, we find the northward component of the wind using the sine function.

Northward component of wind = 80 mph * sin(70 degrees)

Once we have the components of the wind vector, we can add it to the components of the plane's velocity vector (320 mph, aimed 40 degrees N of E) to find the resulting ground speed and direction.

To add these vectors, we add their x-components together and their y-components together.

Eastward component = 320 mph * cos(40 degrees)
Northward component = 320 mph * sin(40 degrees)

Now, we add the components:

Resultant eastward component = Eastward component of wind + Eastward component of plane's velocity
Resultant northward component = Northward component of wind + Northward component of plane's velocity

Lastly, we can calculate the magnitude (ground speed) and direction (angle) of the resulting vector using the Pythagorean theorem and trigonometry.

Magnitude (ground speed) = sqrt((Resultant eastward component)^2 + (Resultant northward component)^2)

Direction (angle) = arctan(Resultant eastward component / Resultant northward component)

Plug in the calculated values to determine the resulting ground speed and the angle at which the plane should aim to fly in the given wind conditions.