A scout troop is practicing its orienteering skills with map and compass. First they walk due east for 1.5 km. Next, they walk 32° west of north for 3.5 km. How far and in what direction must they walk to go directly back to their starting point?
Check previous post for solution.
To determine how far and in what direction the scout troop must walk to go directly back to their starting point, we can break down their movements into east-west and north-south components.
First, let's consider their east-west movement. They initially walk due east, which means they have moved 1.5 km eastward. To go back to their starting point in the east-west direction, they need to move 1.5 km westward.
Next, let's consider their north-south movement. They move 32° west of north. This means they have deviated 32° to the left from the north direction. To determine the north-south component of this movement, we need to find the opposite side of this angle, which is the sine component. We can use trigonometry to find this component.
The north-south component can be calculated by multiplying the distance (3.5 km) by the sine of the angle (32°):
North-South component = 3.5 km * sin(32°) = 1.83 km
To go back to their starting point in the north-south direction, they need to move 1.83 km southward.
Finally, we can combine the east-west and north-south components to find the total distance and direction they must walk. We use the Pythagorean theorem to calculate the total distance:
Total distance = √(east-west component^2 + north-south component^2)
= √(1.5 km^2 + 1.83 km^2)
= √(2.25 km^2 + 3.3489 km^2)
= √5.5989 km^2
= 2.37 km (approximately)
To determine the direction, we can use trigonometry again. The angle can be calculated by taking the inverse tangent of the east-west component divided by the north-south component:
Angle = tan^(-1)(east-west component / north-south component)
= tan^(-1)(1.5 km / 1.83 km)
= tan^(-1)(0.81967)
= 39.8° (approximately)
Therefore, the scout troop must walk approximately 2.37 km at an angle of 39.8° (measured counterclockwise from due north) to go directly back to their starting point.