The angles of elevation to an airplane from two points A and B on level ground are 55° and 72°, respectively. The points A and B are 2.8 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.)
(a)
Let point A be at position O and point B be at position P.
Let the airplane be at position X.
Draw a horizontal line OP, with A at one end and B at the other.
Draw lines from A and B upwards at angles of 55° and 72° respectively to meet at point X.
Label the distance between points A and B as 2.8 miles.
(b)
Let the distance between the plane and point B be h miles.
In triangle PBX, we have
tan(72°) = h / 2.8
h = 2.8 * tan(72°)
h ≈ 6.99 miles
So, the distance between the plane and point B is approximately 6.99 miles.
(c)
Let the altitude of the plane be y miles.
In triangle PBX, we have
tan(72°) = y / h
y = h * tan(72°)
y ≈ 6.99 * tan(72°)
y ≈ 19.02 miles
So, the altitude of the plane is approximately 19.02 miles.
(d)
Let the distance the plane must travel before it is directly above point A be d miles.
In triangle PAX, we have
tan(55°) = y / d
d = y / tan(55°)
d ≈ 19.02 / tan(55°)
d ≈ 13.40 miles
So, the distance the plane must travel before it is directly above point A is approximately 13.40 miles.