Instructions for finding a buried treasure include the following: Go 73.0 paces at 234°, turn to 135° and walk 122 paces, then travel 120 paces at 167°. The angles are measured counterclockwise from an axis pointing to the east, the + x direction. Determine the resultant displacement from the starting point. Enter the distance in paces (without units) and the angle relative to the positive x-axis
To determine the resultant displacement from the starting point, we need to find the vector sum of each individual displacement. Let's break down the instructions into each step and calculate the resulting displacement.
Step 1: Go 73.0 paces at 234°
To calculate the horizontal and vertical components of this displacement, we can use trigonometry. Let's consider the angle as measured counterclockwise from the positive x-axis.
Horizontal component: 73.0 paces * cos(234°) = -43.845 paces (since the angle is 234°, it's in the third quadrant, which is in the negative x-direction)
Vertical component: 73.0 paces * sin(234°) = -43.845 paces (since the angle is 234°, it's in the third quadrant, which is in the negative y-direction)
Step 2: Turn to 135° and walk 122 paces
At this point, we need to add this displacement to the previous result. Again, considering the angle counterclockwise from the positive x-axis.
Horizontal component: 122 paces * cos(135°) = -86.375 paces (since the angle is 135°, it's in the second quadrant, which is in the negative x-direction)
Vertical component: 122 paces * sin(135°) = 86.375 paces (since the angle is 135°, it's in the second quadrant, which is in the positive y-direction)
Step 3: Travel 120 paces at 167°
We'll again add this displacement to the previous result, considering the angle counterclockwise from the positive x-axis.
Horizontal component: 120 paces * cos(167°) = -111.635 paces (since the angle is 167°, it's in the third quadrant, which is in the negative x-direction)
Vertical component: 120 paces * sin(167°) = -58.898 paces (since the angle is 167°, it's in the third quadrant, which is in the negative y-direction)
Now, we'll calculate the resultant horizontal and vertical components by adding up all the individual components:
Horizontal component = -43.845 paces + (-86.375 paces) + (-111.635 paces) = -241.855 paces
Vertical component = -43.845 paces + (86.375 paces) + (-58.898 paces) = -16.368 paces
Finally, we'll use these components to calculate the magnitude (distance) and angle relative to the positive x-axis:
Magnitude: sqrt((-241.855 paces)^2 + (-16.368 paces)^2) ≈ 242.19 paces (rounded to two decimal places)
Angle: atan((-16.368 paces) / (-241.855 paces)) ≈ 4.57° (rounded to two decimal places)
Therefore, the resultant displacement from the starting point is approximately 242.19 paces at an angle of 4.57° relative to the positive x-axis.