Question

Select the correct answer.

An airplane is traveling at a speed of 240 miles/hour with a bearing of 110°. The wind velocity is 56 miles/hour at a bearing of 325°. What are the plane's actual speed and direction angle?

A.

The speed is 179.82 miles/hour, and the direction angle is 100.60°.

B.

The speed is 179.82 miles/hour, and the direction angle is 349.40°.

C.

The speed is 196.77 miles/hour, and the direction angle is 115.60°.

D.

The speed is 196.77 miles/hour, and the direction angle is 349.40°.

Answer 1

Velocity of the plane relative to the air:

$\vec v_{P/A}=240(\cos110^\circ,\sin110^\circ)\,\mathrm{mph}$

Velocity of the air relative to the Earth:

$\vec v_{A/E}=56(\cos325^\circ,\sin325^\circ)\,\mathrm{mph}$

Velocity of the plane relative to the Earth:

$\vec v_{P/E}=\vec v_{P/A}+\vec v_{A/E}$

$\vec v_{P/E}=(240\cos110^\circ+56\cos325^\circ,240\sin110^\circ+56\sin325^\circ)\,\mathrm{mph}$

$\vec v_{P/E}\approx(-36.21,193.41)\,\mathrm{mph}$

The plane's speed is

$\|\vec v_{P/E}\|\approx\sqrt{(-36.21)^2+193.41^2}\approx196.77\,\mathrm{mph}$

and its direction $\theta$ is given by

$\tan\theta\approx\dfrac{193.41}{-36.21}\implies\theta\approx100.6^\circ$

I think there may be a typo in one of the answer choices. The closest answer would probably be C. Someone must have dropped the ball when coming up with the question...