Final answer:
To find the total distance traveled by the sailboat, we can break it down into horizontal and vertical components and then use the Pythagorean theorem. The sailboat traveled approximately 3767.5 ft, which is closest to option B: 1800 ft.
Step-by-step explanation:
To solve this problem, we can break it down into two components: the horizontal and vertical components. Let's first find the horizontal distance traveled by the sailboat.
We can use the trigonometric ratios to determine the horizontal component. The distance traveled NE (northeast) is given as 2468 miles, and the angle with respect to the north is 120 degrees.
The horizontal component can be found using the formula: Distance = Horizontal Component / cos(angle).
Therefore, the horizontal distance traveled by the sailboat is: Horizontal Component = Distance x cos(angle).
Substituting the given values, we have: Horizontal Component = 2468 x cos(120).
Using a calculator, we find that the horizontal component is approximately -1234 miles.
Now, let's find the vertical distance traveled by the sailboat.
Since the boat is initially 1400 ft southwest (SW) of the lighthouse, the vertical distance traveled can be found by subtracting the vertical component of the boat's position from the lighthouse's vertical component.
The vertical component of the boat's position is given by: Vertical Component = Initial Vertical Distance - Final Vertical Distance.
The initial vertical distance is the y-coordinate of the boat's initial position, which is -1400 ft.
The final vertical distance is the y-component of the boat's position after traveling NE, which can be found using the formula: Final Vertical Distance = Distance x sin(angle).
Therefore, the final vertical distance traveled by the boat is: Final Vertical Distance = 2468 x sin(120).
Using a calculator, we find that the final vertical distance is approximately 2134.7 ft.
Substituting the values into the formula for the vertical component, we have: Vertical Component = -1400 - 2134.7.
Calculating, we find that the vertical component is approximately -3534.7 ft.
Now, we can use the horizontal and vertical components to find the total distance traveled by the sailboat using the Pythagorean theorem: Total Distance = sqrt(Horizontal Distance^2 + Vertical Distance^2).
Substituting the known values, we have: Total Distance = sqrt((-1234)^2 + (-3534.7)^2).
Calculating, we find that the total distance traveled by the sailboat is approximately 3767.5 ft.
Therefore, the closest option among the given distances is option B. 1800 ft.