An airplane is travelling at a speed of 600 mph on a heading of 345 degrees. There is a wind moving at a speed of

70 mph on a heading of 20 degrees. The pilot would like to maintain his planned course. Find the speed and heading he should set in order to travel with the wind and keep on course.

so his his heading + wind=345

let Theta be his heading steered.
600sinTheta W + 600cosTheta E + 70 sin20 E + 70 cos 20 N=SOG*sin345 W + SOG cos345 N
(SOG is speed over grund)

set the components equal
West: 600 sinTheta W+70cos20 E=sog sin345 W
or
change the second direction E to -W, solve for Theta.
Put that into the N equation to solve for SOG.

To find the speed and heading the pilot should set in order to travel with the wind and stay on course, we need to calculate the resultant velocity of the airplane in relation to the ground.

First, we need to break down the velocities of the airplane and the wind into their horizontal and vertical components. We can use trigonometry for this.

The horizontal component of the airplane's velocity can be found using the formula:

Horizontal component = Airplane speed * cos(Airplane heading)

So, the horizontal component of the airplane's velocity is:

Horizontal component of airplane velocity = 600 mph * cos(345 degrees)

Similarly, the vertical component of the airplane's velocity can be found using the formula:

Vertical component = Airplane speed * sin(Airplane heading)

So, the vertical component of the airplane's velocity is:

Vertical component of airplane velocity = 600 mph * sin(345 degrees)

Now, let's do the same calculations for the wind velocity:

Horizontal component of wind velocity = 70 mph * cos(20 degrees)
Vertical component of wind velocity = 70 mph * sin(20 degrees)

Next, we need to sum up the horizontal components of the airplane and wind velocities to get the resultant horizontal velocity:

Resultant horizontal velocity = Horizontal component of airplane velocity + Horizontal component of wind velocity

Similarly, we sum up the vertical components of the airplane and wind velocities to get the resultant vertical velocity:

Resultant vertical velocity = Vertical component of airplane velocity + Vertical component of wind velocity

Now, we can use the Pythagorean theorem to find the magnitude of the resultant velocity. The formula is:

Resultant velocity = √(Resultant horizontal velocity^2 + Resultant vertical velocity^2)

Finally, we can use trigonometry to find the heading (direction) of the resultant velocity. The formula is:

Resultant heading = arctan(Resultant vertical velocity / Resultant horizontal velocity)

These calculations will give us the speed and heading the pilot should set in order to travel with the wind and stay on course.