A boat travels 9 km due south and then 7 km due west. What angle should the boat make with the north line to return to the base (see figure to the right)?
To find the angle that the boat should make with the north line to return to the base, we can use trigonometry.
First, let's draw a diagram for better visualization. Since the boat travels 9 km due south and then 7 km due west, we can create a right-angled triangle. Let's label the sides of the triangle accordingly:
|
|
| 9 km
A |
|
-------+-------
|
| B
|
| 7 km
|
|
|
In this triangle, side A represents the displacement to the south, which is 9 km. Side B represents the displacement to the west, which is 7 km. We want to find the angle between the north line and the boat's path.
Now, we can use the trigonometric function tangent (tan) to find the angle. The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side.
In this case, the angle we're looking for is the one opposite side B and adjacent to side A. So, we can use the formula:
tan(angle) = opposite/adjacent
tan(angle) = B/A
Substituting the values we have:
tan(angle) = 7/9
Now, to find the angle itself, we'll take the inverse tangent (or arctan) of both sides of the equation:
angle = arctan(7/9)
Using a scientific calculator or an online trigonometry calculator, you can find the arctan of 7/9 to get the angle. In this case, the angle is approximately 38.66 degrees.
Therefore, the boat should make an angle of approximately 38.66 degrees with the north line to return to the base.
tan (angle W of S) = 7/9
so went outbound 37.9 deg west of south
so
steer 37.9 degrees East of North to get back