Instructions for finding a buried treasure include
the following: Go 585.3 paces at 156◦
,
turn to 215◦
and walk 282 paces, then travel
400 paces at 132◦
.
Find the magnitude of the resultant displacement
from the starting point.
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◦
.
Answer in units of ◦
.
585.3 paces at 156◦
cos 156 = -.914
sin 156 = .407
x distance = 585.3 *- .914 = - 525
y distance = 585.3 * .407 = + 238
282 at 215
cos 215 = -.819
sin 215 = -.574
x distance = 282 * -.819
y distance = 282 * -.574
400 at 132
cos 132 = - ....
sin 132 = + ...
x ....
y ....
Now add all the x distances X = .....
then Y = .....
figure out what quadrant by X and Y signs
tan (angle from x axis) = y/x
find the x and y components of the individual segments , then add
... the sine of the bearing is the x component , the cosine is the y component
(magnitude)^2 = x^2 + y^2
tan(bearing) = x / y
Scott, cos is x, they are using counterclockwise from east, not compass directions. Bothers me too (teach navigation).
oops ... got x and y reversed
sine is y ... cosine is x
tan(bearing) = y / x
and sorry, tan bearing = Y/X
we are used to navigating clockwise from north, but this is counterclockwise from east
But what about the second part? The direction?
As Scott and I both told you
to get the direction take the inverse tangent of Y/X
Be careful about the quadrant. If in quadrant 2 for example angle is between 90 and 180
To find the magnitude of the resultant displacement, you will need to use the concept of vector addition.
First, let's break down the instructions into individual displacement vectors.
1. The first instruction is to go 585.3 paces at 156 degrees.
This gives us a displacement vector of magnitude 585.3 and direction 156 degrees.
2. The second instruction is to turn to 215 degrees and walk 282 paces.
This gives us a displacement vector of magnitude 282 and direction 215 degrees.
3. The third instruction is to travel 400 paces at 132 degrees.
This gives us a displacement vector of magnitude 400 and direction 132 degrees.
Now, let's add these vectors together to find the resultant displacement.
To add vectors, we need to break them down into their horizontal (x) and vertical (y) components, using trigonometry.
For the first instruction:
Horizontal component = 585.3 * cos(156)
Vertical component = 585.3 * sin(156)
For the second instruction:
Horizontal component = 282 * cos(215)
Vertical component = 282 * sin(215)
For the third instruction:
Horizontal component = 400 * cos(132)
Vertical component = 400 * sin(132)
Now, sum up the horizontal and vertical components separately to get the resultant displacement vector:
Horizontal component sum = (585.3 * cos(156)) + (282 * cos(215)) + (400 * cos(132))
Vertical component sum = (585.3 * sin(156)) + (282 * sin(215)) + (400 * sin(132))
Next, calculate the magnitude of the resultant displacement using the Pythagorean theorem:
Magnitude = sqrt((Horizontal component sum)^2 + (Vertical component sum)^2)
Now that you have the magnitude of the resultant displacement, you can find the direction.
To find the direction of the resultant displacement, use the inverse tangent function:
Direction = atan2(Vertical component sum, Horizontal component sum)
Make sure to convert the angle from radians to degrees within the range of -180 degrees to +180 degrees.
Finally, plug in the calculated values into the formula and solve for the magnitude and direction of the resultant displacement.