Final answer:
The correct equation to measure the cyclist's distance from the starting point is x² = 5² + 3² - 2(5)(3)cos(100°), applying the Law of Cosines with the included angle of 100°.
Step-by-step explanation:
To calculate the cyclist's distance x from the starting point, we can use the Law of Cosines. The correct equation which measures the cyclist's distance x from the starting point after traveling 5 miles in the direction N 30° E and then 3 miles in the direction N 70° E is:
x² = 5² + 3² - 2(5)(3)cos(100°)
Here's why: when the cyclist changes direction, the angle between the two legs of the journey is the difference between the two bearings, which is 70° - 30° = 40°. However, since we're looking for the angle outside of the triangle formed by the two legs of the journey and the displacement, we need to subtract this from 180°, giving us 180° - 40° = 140°, but in this case the cosine of the supplemental angle (180° - 140° = 40°) is the same as the cosine of 140°, which means we want the cosine of 100° for our calculation.
Using the Law of Cosines:
x² = a² + b² - 2ab*cos(θ), where a and b are the sides of the triangle adjacent to the angle θ.