Solve the problem.
-A bird flies from its nest $7 \mathrm {~km}$ in the direction $2 ^ { \circ }$ north of east, where it stops to rest on a tree. It then flies $10 \mathrm {~km}$ in the direction $4 ^ { \circ }$ south of west and lands atop a telephone pole. With an xy-coordinate system where the origin is the bird's nest, the $x$-axis points east, and the $y$-axis points north, at what point is the telephone pole located?
A) (6.996, 0.2443)
B) (-9.976, -0.6976)
C) (-2.980, -0.4533)
D) (3.623, 10.53)

Question

Quizplus · Accepted Answer

The bird's movements can be broken down into components along the x (east-west) and y (north-south) axes. For the first leg of the journey, the bird flies 7 km at 2 degrees north of east. The x-component is $7 \cos(2^\circ)$ and the y-component is $7 \sin(2^\circ)$. For the second leg, the bird flies 10 km at 4 degrees south of west. The x-component is $-10 \cos(4^\circ)$ (negative because it's westward) and the y-component is $-10 \sin(4^\circ)$ (negative because it's southward). Adding these components gives the total displacement from the origin. Calculating these, the x-coordinate is approximately $-2.980$ and the y-coordinate is approximately $-0.4533$, placing the telephone pole at $(-2.980, -0.4533)$.

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