An airplane is traveling at a speed of 240 miles/hour with a bearing of 110°. the wind velocity is 56 miles/hour at a bearing of 325°. what are the plane's actual speed and direction angle? a. the speed is 179.82 miles/hour, and the direction angle is 100.60°. b. the speed is 179.82 miles/hour, and the direction angle is 349.40°. c. the speed is 196.77 miles/hour, and the direction angle is 115.60°. d. the speed is 196.77 miles/hour, and the direction angle is 349.40°.

  none of the above

Step-by-step explanation:

Two different calculators show the sum of plane velocity and wind velocity to be 196.77 mph at a bearing of 100.60°.

In the triangle of vectors, the side lengths are 240 and 56, and the angle between them is 325° -(110°+180°) = 35°. Then the law of cosines gives the magnitude of the resultant speed (s) as ...

  s^2 = 240^2 +56^2 -2·240·56·cos(35°) ≈ 38717.1930

  s ≈ √38717.1930 ≈ 196.767 . . . . mph

Then the angle difference (d) between the plane's direction and the resultant direction can be found from the law of sines:

  sin(d)/56 = sin(35°)/196.767

  d = arcsin(56/196.767·sin(35°)) ≈ 9.395°

meaning that the actual direction is 110° -9.395° = 100.605° ≈ 100.60°

The plane's actual speed and direction are 196.77 miles/hour at a bearing of 100.60°.

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Comment on bearing angles

Bearing angles are conventionally measured clockwise from north. If you draw your map so that north is to the right (+x direction) and east is up (+y direction), then angles conventionally measured or computed in algebra and trigonometry will be the same as bearing angles. That is, we can use bearing angles in computations without worrying about how they're measured in the real world. We just need to be aware of the relationship they have to north and east.

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Comment on answer choices

We consider it unfortunate that the offered answer selections do not include the actual answer. That happens sometimes. The only thing we can suggest is that you have your teacher work this problem for the class, so you can see how the offered answers relate to that working.

(The actual speed cannot be outside the range 240±56, so cannot be 179.82 miles per hour—eliminating choices A and B. The plane's direction angle cannot be altered by more than arctan(56/240) = 13.13°, so cannot be 349.40°—eliminating choices B and D. Since the direction angle of the remaining choice C is greater than 110°, the required wind direction is less than 180°+110° = 290°. The actual wind bearing of 325° cannot push the plane in that direction.)

true

explanation:

is means division by fractions

step-by-step explanation: