Question

1. a motorcycle traveled 750 miles at 70 degrees south of east. how far east did the motorcycle travel?

○ 194.1 miles east
○ 679.7 miles east
○ 704.8 miles east
○ 256.5 miles east

Explanation:

Answer:

To determine how far east the motorcycle traveled, we use trigonometry. The eastward distance is the adjacent side of the right triangle formed by the motorcycle's path (hypotenuse = 750 miles) and the angle of 70 degrees south of east. The formula for the adjacent side (eastward distance) is \( \text{adjacent} = \text{hypotenuse} \times \cos(\theta) \), where \( \theta = 70^\circ \).

Calculating:
\( \text{Eastward distance} = 750 \times \cos(70^\circ) \)
\( \cos(70^\circ) \approx 0.3420 \)
\( 750 \times 0.3420 \approx 256.5 \)

Wait, no—wait, I made a mistake. Wait, "south of east" means the angle between the east direction and the path is 70 degrees towards the south. So the east component is adjacent, so \( \cos(70^\circ) \) times hypotenuse? Wait, no, wait: if the angle is south of east, the east component is \( 750 \times \cos(70^\circ) \)? Wait, no, wait, let's visualize: east is the x-axis, south is the negative y-axis. The angle between the path (hypotenuse) and the east (x-axis) is 70 degrees below the x-axis. So the east component (x-component) is \( 750 \times \cos(70^\circ) \), and the south component is \( 750 \times \sin(70^\circ) \). Wait, but when I calculate \( 750 \times \cos(70^\circ) \):

\( \cos(70^\circ) \approx 0.3420 \), so \( 750 \times 0.3420 = 256.5 \). But wait, one of the options is 256.5 miles east. Wait, but let me check again. Wait, maybe I mixed up sine and cosine? Wait, no: in standard position, east is 0 degrees, south is 270 degrees, but "south of east" is 70 degrees below the east (x-axis) direction. So the angle with the east axis is 70 degrees, so the east component is adjacent, so cosine. So \( 750 \times \cos(70^\circ) \approx 750 \times 0.3420 = 256.5 \). So the correct answer is 256.5 miles east.

Wait, but let me verify with a calculator. \( \cos(70^\circ) \approx 0.3420201433 \). Then \( 750 \times 0.3420201433 \approx 256.5151075 \), which rounds to 256.5 miles east. So the correct option is "256.5 miles east".