The angles of elevation theta and alpha to an airplane from the airport control tower and from an observation post 2 miles away are being continuously monitored. If theta is 42 degrees when alpha is 81 degrees, how far is the plane from the observation post?
You need to clarify the diagram. Label the airplane A, the tower T and the post P. The plane is at altitude h, with AP=t and AT=p. Point Q is directly below the plane. The distance PQ=q and thus TQ = 2-q.
If the plane is between the tower and the post, then
h/q = tanα
h/(2-q) = tanθ
Eliminate h and plug in the angles, and to get the desired distance t,
q tan 81° = (2-q)tan42°
6.31q = 0.90(2-q)
q = 0.249
so, h = 0.249*6.31 = 1.571
t^2 = .249^2 + 1.571^2
t = 1.59 miles
If the plane is not between the two measurement locations, then you need to set up different triangles.
To solve this problem, we can use the concept of trigonometry and set up a right triangle. Let's call the distance from the observation post to the airplane "x" (in miles).
We have two angles, theta (angle of elevation from the control tower) and alpha (angle of elevation from the observation post). We are given that theta is 42 degrees when alpha is 81 degrees.
In a right triangle, the tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. Using this relationship, we can set up equations:
From the control tower:
tan(theta) = height of the airplane / distance from the control tower
tan(42) = h / 2
From the observation post:
tan(alpha) = height of the airplane / distance from the observation post
tan(81) = h / (2 + x)
Where h represents the height of the airplane.
Now, we can solve these equations to find the value of x.
First, let's find the value of h by equating the two equations above:
tan(42) = tan(81) * (2 + x) / 2
Multiply both sides by 2:
2 * tan(42) = tan(81) * (2 + x)
Now, divide both sides by tan(81):
2 * tan(42) / tan(81) = 2 + x
Subtract 2 from both sides:
2 * tan(42) / tan(81) - 2 = x
Now we can calculate this value using a scientific calculator:
x ≈ 1.339
Therefore, the plane is approximately 1.339 miles away from the observation post.