A cruise ship travels 310 miles due east before turning 20 degrees north of east it travels 150 miles along its new course. How far is the cruise ship from its initial position?
To find the distance from the ship's initial position, we can break the 150-mile journey into its eastward and northward components.
The eastward component is 150 * cos(20 degrees) miles and the northward component is 150 * sin(20 degrees) miles.
150 * cos(20 degrees) = 141.46 miles eastward
150 * sin(20 degrees) = 51.65 miles northward
Now, we can use the Pythagorean theorem to find the distance from the ship's initial position:
Distance = sqrt((310 + 141.46)^2 + (51.65)^2)
Distance ≈ sqrt((451.46)^2 + (51.65)^2)
Distance ≈ sqrt(203762 + 2668.3)
Distance ≈ sqrt(206430.3)
Distance ≈ 454.40 miles
Therefore, the cruise ship is approximately 454.40 miles from its initial position.