I really need help... You are on a treasure hunt and your map says "Walk due west for 52 paces, then walk 29.0° north of west for 39 paces, and finally walk due north for 25 paces." What is the magnitude of the component of your displacement in the following directions?
Thank You
To find the magnitude of the component of your displacement in different directions, we can break down the given instructions into the x-axis (east-west) and y-axis (north-south) components.
1. Walking due west for 52 paces:
The displacement in the x-axis (east-west) component is -52 paces, where the negative sign indicates walking west.
2. Walking 29.0° north of west for 39 paces:
To find the x-axis (east-west) component, we need to calculate the cosine of the angle. Given that the angle is north of west, we subtract 180° from it to get 29.0° - 180° = -151.0°.
Now, we can use the cosine function: cos(-151.0°) = -0.774.
The displacement in the x-axis (east-west) component is then -0.774 * 39 = -30.186.
To find the y-axis (north-south) component, we need to calculate the sine of the angle.
sin(-151.0°) = -0.634.
The displacement in the y-axis (north-south) component is -0.634 * 39 = -24.726.
3. Walking due north for 25 paces:
The displacement in the y-axis (north-south) component is 25 paces.
Now, to find the magnitude of each component, we can use the Pythagorean theorem.
Magnitude of the x-axis component: sqrt((-52)^2 + (-30.186)^2) ≈ 60.13 paces.
Magnitude of the y-axis component: sqrt((-24.726 + 25)^2) = 0.274 paces.
So, the magnitude of the component of your displacement in the east-west (x-axis) direction is approximately 60.13 paces, and in the north-south (y-axis) direction is 0.274 paces.
29 deg N of W = 151 deg(180 - 29) CCW from 0.
X = hor = -52 + 39cos151 = -52 - 34 = -86,
Y = ver = 39sin151 + 25 = 18.9 + 25 = 43.9,
D^2 = X^2 + Y^2 = (86)^2 + (43.9)^2 = 9331.6,
D = sqrt(9331.6) = 96.6.