The angles of elevation to an airplane from two points A and B on level ground are 54° and 72°, respectively. The points A and B are 2.5 miles apart, and the airplane is east of both points in the same vertical plane.
(a) Draw a diagram that represents the problem. Show the known quantities on the diagram.
(b) Find the distance between the plane and point B. (Round your answer to two decimal places.)
(c) Find the altitude of the plane. (Round your answer to two decimal places.)
(d) Find the distance the plane must travel before it is directly above point A. (Round your answer to two decimal places.)
(c) If the plane is at height h, then
h cot54° - h cot 72° = 2.5
h = 6.22
(b) if the distance from plane to B is x, then
h/x = sin72°
x = 6.54
(d) 2.5 + 6.22 cot72° = 4.52
(a)
A (54°) B (72°)
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P (airplane)
Known quantities:
Angle of elevation at Point A = 54°
Angle of elevation at Point B = 72°
Distance between points A and B (AB) = 2.5 miles
(b) Let x be the distance between the plane and point B. Using trigonometry:
tan(72°) = x / 2.5
x = 2.5 * tan(72°)
x ≈ 6.32 miles
The distance between the plane and point B is approximately 6.32 miles.
(c) Let h be the altitude of the plane. Using trigonometry:
tan(54°) = h / 2.5
h = 2.5 * tan(54°)
h ≈ 3.25 miles
The altitude of the plane is approximately 3.25 miles.
(d) Let d be the distance the plane must travel before it is directly above point A. Using trigonometry:
tan(54°) = h / d
d = h / tan(54°)
d ≈ 3.25 / tan(54°)
d ≈ 2.66 miles
The distance the plane must travel before it is directly above point A is approximately 2.66 miles.
Apologies for the miscalculations in the previous solution. Thank you for providing the corrected calculations. The correct values should be:
(b) Using trigonometry:
If the distance from the plane to point B is x:
h/x = sin(72°)
x = h / sin(72°) = 3.745 / 0.951 = 3.94 miles
The distance between the plane and point B is approximately 3.94 miles.
(c) Using trigonometric identities:
If the plane is at height h:
h cot(54°) - h cot(72°) = 2.5
h * (1/tan(54°) - 1/tan(72°)) = 2.5
h = 2.5 / (1/tan(54°) - 1/tan(72°))
h ≈ 6.22 miles
The altitude of the plane is approximately 6.22 miles.
(d) If d is the distance the plane must travel before it is directly above point A:
d = 2.5 + h cot(72°)
d = 2.5 + 6.22 * 0.37633
d ≈ 4.52 miles
The distance the plane must travel before it is directly above point A is approximately 4.52 miles.