A commuter airplane starts from an airport located at the origin. First it ?ies
ID: 1951155 • Letter: A
Question
A commuter airplane starts from an airportlocated at the origin. First it ?ies to city
A located 96 km away from the airport in
a direction 26
?
North of East. Next it ?ies
59.5 km at 32
? West of North to city B. Finally
it ?ies 153 km due West to city C.
What is the x-component of vector ~r from
the starting point (position O) to the city C?
What is the y-component of vector ~r?
Answer in units of km.
What is the direction of the ?nal position
vector ~r in degrees West of North; i.e., what
is the angle ?? Use counterclockwise as the
positive angular direction, measured between
-180
?and +180
?
from North; (e.g., East is
-90
?
).
Answer in units of
?
Explanation / Answer
The final x component vector is the sum of all of the x component vectors. - for the first segment (96 km at 26°NofE) the x component is 96*cos(26) = 86.28 km - for the second segment (59.5 km at 32° WofN) the x component is -59.5*sin(32) = 31.53 km - for the last segment (153 W) the x component is -153 km By adding these three values, you obtain x = -98.25 km The y component vector can be obtained the same way: it is the sum of the y component vectors. - for the first segment (96 km at 26°NofE) the y component is 96*sin(26) = 42.08km - for the second segment (59.5 km at 32° WofN) the y component is 59.5*cos(32) = 50.46km - for the last segment (153 W) the y component is 0 km By adding these three values, you obtain y = 92.54 km The length of the vector can be obtained using the Pythagorean theorem: x^2 + y^2 = r^2. - doing this you obtain r = 134,97 km The angle can be obtained by using tangent. tan(angle) = x/y, so inversetan(x/y) = angle: - inversetan(-98.25/92.54)= -46,71° So your final answer is 134,97 km at 46.71° West of North
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