Solve the problem.
-A plane is heading due south with an airspeed of $223 \mathrm { mph }$. A wind from a direction of $60.0 ^ { \circ }$ is blowing at $18.0$ mph. Find the bearing of the plane. (Note that bearings are measured from north, clockwise.) Round results to an appropriate number of significant digits.
A) $184 ^ { \circ }$
B) $179 ^ { \circ }$
C) $86 ^ { \circ }$
D) $91 ^ { \circ }$

Question

Quizplus · Accepted Answer

To solve this problem, we need to calculate the resultant velocity of the plane considering both its airspeed and the wind's effect. The wind's direction at $60^\circ$ (from north) means it's blowing from the northeast towards the southwest. Since the plane is heading due south, the wind will push it slightly off course to the west.1. Decompose the wind's velocity into north-south and east-west components. The north-south component is $18 \cos(60^\circ) = 9 \, \text{mph}$ south, and the east-west component is $18 \sin(60^\circ) = 15.6 \, \text{mph}$ west.2. The plane's southward velocity is $223 \, \text{mph}$ (its airspeed), and it has an additional 9 mph southward component from the wind, totaling $232 \, \text{mph}$ south.3. The westward component of the wind (15.6 mph) is the only horizontal component affecting the plane's direction.4. To find the bearing, calculate the angle $\theta$ from due south that the resultant velocity makes due to the westward wind component. Use the arctan of the ratio of the westward component to the total southward component: $\theta = \arctan\left(\frac{15.6}{232}\right)$.5. $\theta \approx \arctan\left(\frac{15.6}{232}\right) \approx \arctan\left(0.0672\right) \approx 3.84^\circ$ west of south.6. Since bearings are measured clockwise from north, the bearing of the plane is $180^\circ + 3.84^\circ = 183.84^\circ$, which rounds to $184^\circ$.Therefore, the correct answer is $184^\circ$, which corresponds to choice A.

Solve the Problem 223mph223 \mathrm { mph }223mph a Wind from a Direction Of Due

Solve the Problem $223 \mathrm { mph }$ a Wind from a Direction Of Due