An explorer in the dense jungles of equatorial Africa leaves his hut. He takes 39 steps at an angle 45 north of east, then 80 steps at an angle 60 north of west, then 51 steps due south. Assume his steps all have equal length. Save him from becoming hopelessly lost in the jungle by giving him the displacement, calculated using the method of components, that will return him to his hut.
To solve this problem using the method of components, we need to break down the explorer's movements into their horizontal (x-axis) and vertical (y-axis) components.
Step 1: Analyze the 39 steps at an angle of 45 degrees north of east:
- Since the angle is north of east, it means the angle is in the northeastern direction.
- To find the horizontal component (x-axis), we need to find the cosine of the angle and multiply it by the number of steps. So, the horizontal component is 39 * cos(45°).
- To find the vertical component (y-axis), we need to find the sine of the angle and multiply it by the number of steps. So, the vertical component is 39 * sin(45°).
Step 2: Analyze the 80 steps at an angle of 60 degrees north of west:
- Since the angle is north of west, it means the angle is in the northwestern direction.
- To find the horizontal component (x-axis), we need to find the cosine of the angle and multiply it by the number of steps. So, the horizontal component is -80 * cos(60°). Note the negative sign because the angle is in the opposite direction on the x-axis.
- To find the vertical component (y-axis), we need to find the sine of the angle and multiply it by the number of steps. So, the vertical component is 80 * sin(60°).
Step 3: Analyze the 51 steps due south:
- Since the explorer walks due south, there is no horizontal (x-axis) component, as he is not moving horizontally.
- To find the vertical component (y-axis), we just need to consider the number of steps taken south, which is -51 because it's in the downward direction.
Now, let's calculate the horizontal and vertical components and find the overall displacement:
Horizontal Component (x-axis):
= 39 * cos(45°) + (-80 * cos(60°))
= 27.73 - (-40)
Vertical Component (y-axis):
= 39 * sin(45°) + 80 * sin(60°) - 51
= 27.73 + 69.28 - 51
Overall Displacement:
= √((Horizontal Component)^2 + (Vertical Component)^2)
= √(27.73^2 + (27.73 + 69.28 - 51)^2)
= √(766.90 + 88.78)
= √855.68
= 29.27
Therefore, the explorer needs to travel approximately 29.27 steps in the direction opposite to the overall displacement to return to his hut and prevent getting lost in the jungle.