Answer: Choice A) 180 km/h [E 14 degrees N]
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Step-by-step explanation:
p = vector for the plane
p = <150,0> is in the form <x,y>, which is cartesian or rectangular form
w = vector for the wind
w = <x,y> is unknown for now
See figure 1 in the attached images. Saying "30 degrees east of north" means that you start looking directly north and then turn 30 degrees toward the east. Based on the diagram attached, the angle between the two vectors p and w is 90-30 = 60. So we will use theta = 60. In figure 1, theta = 60 is the angle BAC. The 30 degree angle in purple is angle CAD. Angles BAC and CAD add to 90 degrees.
The speed of the wind is 50, so the magnitude of vector w is r = 50.
Let's find the x and y components for the wind vector w
x = r*cos(theta)
x = 50*cos(60)
x = 50*0.5
x = 25
y = r*sin(theta)
y = 50*sin(60)
y = 50*0.866025
y = 43.30125
Therefore, w = <25, 43.30125> approximately
We use this to find the resultant vector v. Add up the vectors. Simply add the corresponding x components together. Do the same for y
v = p+w
v = <150,0>+<25,43.30125>
v = <150+25,0+43.30125>
v = <175,43.30125>
We know the resultant vector's cartesian <x,y> form.
Let's find the polar (r,theta) form
r = sqrt(x^2+y^2)
r = sqrt(175^2+43.30125^2)
r = 180.27756
r = 180 <<<--- rounding to the nearest whole number
with the wind factored in, the plane is traveling roughly 180 km/hr. This is the first part of the answer
theta = arctan(y/x) <<<--- arctan is equivalent to inverse tangent
theta = arctan(43.30125/175)
theta = 13.89788 <<<--- make sure you are in degree mode
theta = 14 <<<--- rounding to the nearest whole number
This tells us the plane's direction. The standard position has the angle adjacent to the positive x axis. So we start off facing east, then turn 14 degrees in the north direction. We say that the direction is "14 degrees north of east" which we can write "E 14 N". This is the second part of the answer. The first letter "E" is the first direction we face, which is east. The second letter "N" is the second direction we turn to (north). See figure 2. Hopefully this clears up any confusion you may have about this kind of direction notation.
Side Note: as I finish up, I realize that you can probably skip the direction/angle part and stop when you find r = 180. Why? Because the only answer choice with a speed of 180 km/hr is choice A. So you can effectively eliminate B,C, & D (those choices have speeds of 195, 200, and 132 respectively). Though its always a good idea to know how to answer the full question and not rely on multiple choice answers to eliminate.