A UFO is first sighted at point A due east from an observer at an angle of 14 degrees from the ground. And at an altitude of 223 m. The UFO is next sighted at point B due east at an angle of 28 degrees and an altitude of 446 m. What is the distance from A to B? Round your answer to 3 decimal places.
To find the distance from point A to point B, we can use trigonometry and the concept of similar triangles.
Let's start by drawing a diagram to visualize the situation. We have an observer at point O and the UFO is sighted at two different locations, points A and B:
O
\
\
\
A
.
.
.
B
We are given the following information:
- The angle of elevation at point A is 14 degrees.
- The altitude at point A is 223 m.
- The angle of elevation at point B is 28 degrees.
- The altitude at point B is 446 m.
Now, we can see that triangles OAB and OBA are similar triangles since both have a right angle and share angles A and B. Therefore, we can set up the following proportion based on the corresponding sides:
(OA)/(OB) = (AB)/(OA)
Let's solve for OA (the distance from point O to point A):
(OA)^2 = (AB)(OB)
OA = sqrt((AB)(OB))
To find AB, we'll use trigonometry. In triangle OAB, the opposite side of angle A is AB and the adjacent side is OA. We can use the tangent function:
tan(A) = (opposite side)/(adjacent side)
tan(14°) = AB/OA
AB = (tan(14°))(OA)
Next, let's calculate OA:
OA = sqrt((AB)(OB))
OA = sqrt((tan(14°))(OA))(OB)
OA^2 = (tan(14°))(OA)(OB)
OA = (tan(14°))(OB)
Now, substitute the value of OA in terms of OB:
(tan(14°))(OB) = (tan(14°))(OA)
(tan(14°))(OB) = (tan(14°))(OB)
We can cancel out the common factor of (tan(14°)) on both sides:
OB = OB
This shows that the distance from A to B, AB, is equal to the distance from the observer at point O to point B, which is OB.
Therefore, the distance from A to B is equal to the altitude at point B, which is 446 m.
So, the distance from A to B is 446 meters.