A pilot is flying to a destination 700 miles away at a bearing of 105 degrees. The cruising speed of the plane is 500 mile per hour and the the wind is blowing between 20 and 30 miles per hour at a bearing of 30 degrees. At what bearing should the pilot aim the plan to compensate for the wind?

I made a diagram of a parallogram, then drew in the diagonal to show the resultant, called it R

Looking at a triange, I have sides 20 and 500 with a contained angle of 105°
R^2 = 20^2 + 500^2 - 2(20)(500)cos105°
R = 505.546 mph
repeating for the side at 30
R2 = 508.59 mph

for first case: R = 505.546
sinØ/500 = sin105/505.546
Ø = 72.8°
so bearing = 72.8 - 30 = 42.8°

for 2nd case R2 = 508.59
sinØ/500 = sin105/508.59
Ø = 71.7
so bearing = 71.7 - 30
= 41.7

his bearing should be between 41.7° and 42.8°

To determine the bearing the pilot should aim the plane to compensate for the wind, we need to calculate the resultant velocity of the plane.

Let's break down the given information:
- Distance to the destination: 700 miles
- Bearing of the destination: 105 degrees
- Plane's cruising speed: 500 miles per hour

Now, we need to consider the effect of the wind. The wind is blowing at a bearing of 30 degrees and a speed between 20 and 30 miles per hour. To simplify the calculation, let's assume the wind speed is the average of the given range, which is 25 miles per hour.

To calculate the resultant velocity, we can use vector addition by considering the plane's velocity and the wind's velocity separately.

The plane's velocity can be converted to a vector by multiplying the speed by the direction cosines. Let's assume the x-axis points north and the y-axis points west (as this alignment is common in aviation).

The plane's velocity vector would be: (500 * cos(105), 500 * sin(105))
= (-320.73, 488.42) miles per hour

Similarly, the wind's velocity vector can be calculated using the given speed and bearing.
The wind's velocity vector would be: (25 * cos(30), 25 * sin(30))
= (21.65, 12.50) miles per hour

Now, we add both vectors together to find the resultant velocity:
Resultant velocity vector = Plane's velocity vector + Wind's velocity vector
= (-320.73 + 21.65, 488.42 + 12.50)
= (-299.08, 500.92) miles per hour

The resultant velocity vector tells us the actual direction and speed of the plane relative to the ground. To find the bearing the pilot should aim the plane to compensate for the wind, we need to calculate the angle made by the resultant velocity vector with respect to the north (x-axis).

The desired bearing can be found using the arctan function:
Bearing = arctan(resultant velocity in y-axis/resultant velocity in x-axis)

Bearing = arctan(500.92/-299.08)

Using a scientific calculator or online calculator, we can find the arctan of (-299.08/500.92) to be approximately -31.8 degrees.

Since the arctan function only returns angles between -90 degrees and 90 degrees, we need to adjust the calculated bearing to get a positive angle. Adding 180 degrees to the result will give us:

Bearing = -31.8 + 180
= 148.2 degrees

Therefore, the pilot should aim the plane at a bearing of 148.2 degrees to compensate for the wind.