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Expert: - 4/24/2009


Question
A DC-10 jumbo jet maintains an airspeed of 600 miles per hour in a southeasterly direction. The velocity of the jet stream is a constant 50 miles per hour from the west. Find the actual speed and direction of the aircraft. I thought maybe you divided 600 by 50 to get the speed, but I'm not sure. That's probably wrong.

Answer

Vectors
Questioner:   brittney
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  pre-cal vectors
Question:  A DC-10 jumbo jet maintains an airspeed of 600 miles per hour in a southeasterly direction. The velocity of the jet stream is a constant 50 miles per hour from the west. Find the actual speed and direction of the aircraft. I thought maybe you divided 600 by 50 to get the speed, but I'm not sure. That's probably wrong.
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Hi, Brittney,

600 mph to the SE is a vector with direction angle -45 deg.
50  mph to the E (from the W) is a vector with angle 0 deg.

Now make a triangle OAB, with:

OA = 50
OB = 600
Angle BOA = 45 degrees.
Now you want to solve the triangle, using:

Law of cosines to find AB, the actual speed, and
Law of sines to find angle OAB (actually you want its supplement), which will be an angle of  minus-(something more than 45 degrees)

Write:

AB^2 = OA^2 + OB^2 - 2 OA OB cos(angle BOA)

AB^2 = (50)^2 + (600)^2 - 2 (50)(600) cos(45)

AB^2 = 2500 + 360000 - 60000 (sqrt(2)/2)

AB^2 =        362500 - 30000 (1.414)

My calculator gives

AB = 565.75

Now for the law of sines:

sin OAB    sin 45
-------- = --------  
 600       565.7  


sin OAB = 600 sin 45 / 565.7

OAB = 48.6 degrees (approx)

which means:

565 mph
east 48.6 degrees to the south.

(I think).

Paul Klarreich

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