A cruise ship travels 310 miles due east before turning 20° north of east. It travels 150 miles along its new course. How far is the cruise ship from its initial position?
A triangle is shown. Its horizontal base is labeled 310 miles, and at the right end of the line segment, the line extends with a dashed line. From the right endpoint of the 310-mile segment, another segment and side of the triangle extends up and to the right, and it’s labeled 150 miles. The angle formed between the 150-mile line segment and the extended dashed horizontal line is labeled 20 degrees. A line is drawn that connects the far endpoints of the 310-mile and the 150-mile segments and completes the triangle. It is labeled x.
(1 point)
Responses
295 miles
295 miles
274 miles
274 miles
454 miles
454 miles
160 miles
To find the distance the cruise ship is from its initial position, we can use the Law of Cosines.
Let's call the distance from the end of the 150-mile segment to the initial position of the cruise ship (x) and the angle between the 310-mile segment and x (160 degrees, since the angle north of east is 20 degrees).
Using the Law of Cosines:
x^2 = 310^2 + 150^2 - 2(310)(150)cos(160)
x^2 = 96100 + 22500 - 93000cos(160)
x^2 = 118600 - 93000cos(160)
x^2 ≈ 118600 - 93000(-0.9397)
x^2 ≈ 118600 + 87473.1
x^2 ≈ 205073.1
x ≈ √205073.1
x ≈ 453.4
So, the cruise ship is approximately 453.4 miles from its initial position.
The closest option provided is 454 miles.