9. A boat travels on a course of bearing N37°W for 83 miles. How many miles north and how many miles west has theboat traveled?
draw the diagram. It should be clear that
north: 83 cos37°
west: 83 sin37°
To determine how many miles north and west the boat has traveled, we can use trigonometric concepts.
First, let's define the bearing angle. In this case, the bearing angle is N37°W. This means that the angle between the boat's direction and due north is 37 degrees in a clockwise direction.
To find the distance traveled north, we need to find the component of the total distance in the north direction. We can do this by using the cosine function, since the cosine of an angle in a right triangle gives us the adjacent side divided by the hypotenuse.
The adjacent side in this case is the distance traveled north, and the hypotenuse is the total distance traveled. Let's call the distance traveled north "x" (in miles):
cos(37°) = x / 83 miles
To solve for x, we rearrange the equation:
x = cos(37°) * 83 miles
Calculating this expression, we find:
x ≈ 65.86 miles
So, the boat has traveled approximately 65.86 miles north.
To find the distance traveled west, we need to find the component of the total distance in the west direction. Since the bearing is almost due west (N37°W), the boat does not have a significant southward or upward component to consider.
Therefore, the remaining distance is in the west direction.
To find the distance traveled west, we can use the sine function since the sine of an angle in a right triangle gives us the opposite side divided by the hypotenuse.
Let's call the distance traveled west "y" (in miles):
sin(37°) = y / 83 miles
To solve for y, we rearrange the equation:
y = sin(37°) * 83 miles
Calculating this expression, we find:
y ≈ 49.99 miles
So, the boat has traveled approximately 49.99 miles west.
Thus, the boat has traveled approximately 65.86 miles north and 49.99 miles west.