An airplane flies 137 km due west from city A to city B and then 412 km in the direction of 46.5◦ north of west from city B to city C. What is the distance between city A and
city C?
Relative to city A, in what direction is city C? Answer with respect to due east, with the counter-clockwise direction positive, within the limits of −180◦ to +180◦.
V1 = (137km,180deg.).
V2 = (412km,133.5deg.).
1. X = hor. = 137cos180 + 412cos133.5 =
-137 + (-283.6) = -420.6km.
Y = ver. = 137sin180 + 412sin133.5 =
0 + 298.9 = 298.9km.
d = sqrt((-420.6)^2 + (298.9)^2=516km.
from A to C.
2. tanA = Y/X = 298.9 / -420.6=-0.7107.
A = -35.4 deg.,Cw.
A = -35.4 + 180 = 144.6 deg..CCw. =
Direction from A to C.
To find the distance between city A and city C, we can use the concept of vector addition.
First, let's break down the displacement vectors into their respective components.
The displacement from city A to city B is 137 km due west. Since it is only in the west direction, its x-component will be -137 km and the y-component will be 0 km.
The displacement from city B to city C is 412 km at an angle of 46.5° north of west. To find the x and y components, we can use trigonometry. The x-component will be 412 km multiplied by the cosine of 46.5°, and the y-component will be 412 km multiplied by the sine of 46.5°.
x-component = 412 km * cos(46.5°)
y-component = 412 km * sin(46.5°)
Now we can add the x and y components to get the total displacement from city A to city C.
x-total = -137 km + x-component
y-total = 0 km + y-component
Once we have the x-total and y-total, we can use the Pythagorean theorem to find the distance:
distance = √(x-total)^2 + (y-total)^2
To find the direction of city C relative to city A, we can use the inverse tangent function to calculate the angle between the x-axis and the displacement vector from city A to city C.
angle = arctan(y-total / x-total)
Make sure to convert the angle from radians to degrees and adjust the angle within the limits of -180° to +180°.
Now, plug in the values and perform the calculations to find the distance and angle.