An explorer in the dense jungles of equatorial Africa leaves his hut. He takes 43 steps at an angle 45north 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. What is the magnitude of the displacement that will return the explorer to his hut? What is the direction of the displacement that will return the explorer to his hut?
Thank you!! That helped a lot
To solve this problem using the method of components, we will break down the explorer's steps into their north and east components.
Step 1: Calculate the north and east components for each step.
- The 43 steps at an angle 45° north of east:
- North component = 43 * sin(45°) ≈ 30.40 steps
- East component = 43 * cos(45°) ≈ 30.40 steps
- The 80 steps at an angle 60° north of west:
- North component = 80 * sin(60°) ≈ 69.28 steps
- East component = -80 * cos(60°) ≈ -40.00 steps
- The 51 steps due south:
- North component = -51 steps
- East component = 0 steps
Step 2: Calculate the total north and east components.
- Total north component = 30.40 + 69.28 - 51 = 48.68 steps
- Total east component = 30.40 - 40.00 + 0 = -9.6 steps
Step 3: Calculate the magnitude of the displacement.
- Magnitude of displacement = sqrt((Total north component)^2 + (Total east component)^2)
= sqrt((48.68)^2 + (-9.6)^2)
≈ 49.31 steps
Step 4: Calculate the direction of the displacement.
- Direction = arctan(Total east component / Total north component)
= arctan(-9.6 / 48.68)
≈ -11.18°
Since the direction is given as an angle from the north axis, the negative value means it's 11.18° west of north.
Therefore, to return to his hut, the explorer should move approximately 49.31 steps with a direction 11.18° west of north.
To find the displacement, we can use the method of components. First, let's break down each step into its horizontal and vertical components.
For the first set of steps, the explorer takes 43 steps at an angle of 45 degrees north of east. To find the horizontal and vertical components, we can use trigonometry.
The horizontal component of this step (H1) can be calculated by finding the cosine of the angle and multiplying it by the number of steps:
H1 = cos(45) * 43
The vertical component of this step (V1) can be calculated by finding the sine of the angle and multiplying it by the number of steps:
V1 = sin(45) * 43
Similarly, we can find the horizontal and vertical components for the second set of steps where the angle is 60 degrees north of west:
H2 = cos(60) * 80
V2 = sin(60) * 80
For the final set of steps due south, the horizontal component is 0 since the explorer is moving directly south:
H3 = 0
The vertical component (V3) can be calculated by taking the number of steps directly south, which is 51.
Now, we can add up the horizontal and vertical components separately to find the total displacement.
Horizontal displacement (H) = H1 + H2 + H3 = cos(45) * 43 + cos(60) * 80 + 0
Vertical displacement (V) = V1 + V2 + V3 = sin(45) * 43 + sin(60) * 80 + 51
To find the magnitude of the displacement (D), we can use the Pythagorean theorem:
D = sqrt(H^2 + V^2)
And to find the direction, we can use inverse trigonometry:
θ = arctan(V / H)
Now, let's calculate the values:
Horizontal displacement:
H = cos(45) * 43 + cos(60) * 80 + 0
H ≈ 30.38 + 40 + 0
H ≈ 70.38
Vertical displacement:
V = sin(45) * 43 + sin(60) * 80 + 51
V ≈ 30.38 + 69.28 + 51
V ≈ 150.66
Magnitude of displacement:
D = sqrt(H^2 + V^2)
D = sqrt(70.38^2 + 150.66^2)
D ≈ sqrt(4949.1444 + 22699.3956)
D ≈ sqrt(27648.54)
D ≈ 166.17
Direction of displacement:
θ = arctan(V / H)
θ = arctan(150.66 / 70.38)
θ ≈ arctan(2.1384)
θ ≈ 63.80 degrees north of east
Therefore, the magnitude of the displacement that will return the explorer to his hut is approximately 166.17 steps, and the direction of the displacement is approximately 63.80 degrees north of east.
His first steps lead him to a point 30.4055 steps north and east of his hut.
His second steps lead him to a point 99.687 steps north and 9.5945 steps west of his hut.
His third steps lead him to a point 48.687 steps north and 9.5945 steps from his hut.
Therefore, his hut is d = sqrt(48.687^2 + 9.5945^2) = 49.623 steps at an angle of arctan(9.5945/49.623) = 10.94º east of south.