Instructions for building a buried treasure include the following: Go 872 paces at 167 degrees turn to 286 degrees and walk 135 paces, the travel 319 paces at 152 degrees.
A) Find the magnitude of the resultant displacement from the starting point. B) What is the direction of the resultant displacement? use counterclockwise from due East as the positive angular direction, between the limits of -180 and +180 degrees.
To find the magnitude and direction of the resultant displacement from the starting point, we can use vector addition.
Step 1: Convert the given paces and degrees into vector components.
The first set of instructions are: "Go 872 paces at 167 degrees."
To find the components, we can use trigonometry. The horizontal component (x) can be found by multiplying the magnitude (paces, 872) by the cosine of the angle (167°), and the vertical component (y) can be found by multiplying the magnitude by the sine of the angle.
x₁ = 872 * cos(167°)
y₁ = 872 * sin(167°)
Similarly, for the second set of instructions: "Turn to 286 degrees and walk 135 paces."
x₂ = 135 * cos(286°)
y₂ = 135 * sin(286°)
And for the last set of instructions: "Travel 319 paces at 152 degrees."
x₃ = 319 * cos(152°)
y₃ = 319 * sin(152°)
Step 2: Add up the vector components.
To find the resultant displacement, we add up the x-components and the y-components separately.
x_total = x₁ + x₂ + x₃
y_total = y₁ + y₂ + y₃
Step 3: Find the magnitude and direction.
The magnitude of the resultant displacement can be found using the Pythagorean theorem:
magnitude = sqrt(x_total² + y_total²)
To find the direction, we can use inverse trigonometry functions. Since the question asks for the direction between -180 and +180 degrees counterclockwise from due East, we need to express the result in that range.
direction = atan2(y_total, x_total) * (180 / pi)
if (direction < -180):
direction += 360
elif (direction > 180):
direction -= 360
So, to summarize:
A) Find the magnitude of the resultant displacement from the starting point:
- Calculate the x and y components for each set of instructions.
- Add up the x-components and the y-components.
- Find the magnitude using the Pythagorean theorem: sqrt(x_total² + y_total²).
B) Find the direction of the resultant displacement:
- Use inverse trigonometry functions to find the direction in radians.
- Convert the direction from radians to degrees.
- If the direction is less than -180 degrees, add 360.
- If the direction is greater than 180 degrees, subtract 360.