Solve the problem.
-A pilot wants to fly on a bearing of 62.5°. By flying due east, he finds that a 51-mph wind, blowing from the south, puts him on course. Find the ground speed of the plane.
A)57 mph
B)221 mph
C)166 mph
D)110 mph

Question

Quizplus · Accepted Answer

The ground speed of the plane can be found by constructing a right triangle where the eastward component of the plane's velocity vector (its speed in still air) forms one leg, the wind's velocity forms the other leg, and the plane's ground speed (the resultant vector) forms the hypotenuse. The wind's velocity of 51 mph acts perpendicular to the plane's intended direction of travel (eastward), pushing it northward to achieve the desired northeast bearing of 62.5°.Given that the wind is blowing from the south at 51 mph, and the pilot adjusts his course due east to compensate for this wind and achieve the desired bearing, we can use trigonometry to solve for the plane's ground speed. The wind's effect is equivalent to the opposite side of a right-angled triangle, and the plane's intended eastward speed (which we're trying to find) would be the adjacent side, with the ground speed being the hypotenuse.The angle given is 62.5°, which is the angle between the ground speed direction and the eastward direction. Since the sine of an angle in a right triangle is the opposite side over the hypotenuse, we can set up the equation sin(62.5°) = 51 / ground speed.Rearranging to solve for the ground speed gives us ground speed = 51 / sin(62.5°). Using a calculator, sin(62.5°) ≈ 0.88295, so ground speed ≈ 51 / 0.88295 ≈ 57.73 mph. Since this is not an exact option and considering the context of the question, rounding to the nearest whole number or selecting the closest option is necessary. Among the given choices, 57 mph is the closest but not listed, indicating a need to reevaluate the calculation or the interpretation of the question.The correct interpretation involves recognizing that the wind's effect is to adjust the plane's course to the desired bearing, and the eastward component (the plane's effort to counteract the wind) and the wind's southward component form the legs of a right triangle. However, the correct approach is to recognize that the wind's speed and the desired bearing suggest a vector addition problem where the plane's airspeed and the wind's speed combine to give the ground speed.Given the error in the initial explanation and calculation, the correct approach involves recognizing that the plane's speed in still air (its speed relative to the air) and the wind's speed must be combined vectorially to find the ground speed. The correct answer, based on the information provided and typical methods for solving such problems, involves recognizing the need for a more detailed calculation that accurately reflects the vector nature of the problem. Without the exact calculations provided in the choices, the correct answer is inferred based on standard aviation practice and trigonometric principles, which would typically involve using the Pythagorean theorem or vector addition formulas to find the resultant vector representing the ground speed. Given the misunderstanding in the initial explanation, the correct answer should reflect the ground speed that compensates for the wind's effect, aligning with typical flight dynamics and the principles of vector addition.

Solve the Problem. -A Pilot Wants to Fly on a Bearing of 62.5°