A man at A observes the angle of elevation of a balloon to be 30 degrees.He then walks 1000 meters towards the balloon to a point B and finds the elevation to be 60 degrees. If the balloon has a height of H meters and the man has X meters still to walk before he is directly under the balloon show that H = X tan 60 degrees and H= (X +1000) tan 30 degrees and use these two equations to find X
Let's consider the triangle formed by the balloon, point A, and point B. Let the height of the balloon be H, the distance from point A to point B be X, and the remaining distance from point B to directly under the balloon be 1000 meters.
From the information given, we have two right triangles:
1) In triangle AHB:
- tan(30 degrees) = H/X
- H = X tan(30 degrees)
2) In triangle BHC:
- tan(60 degrees) = H/(X+1000)
- H = (X+1000) tan(60 degrees)
Now, equating the two expressions for H, we have:
X tan(30 degrees) = (X+1000) tan(60 degrees)
Now, we can solve for X:
X tan(30 degrees) = (X+1000) tan(60 degrees)
X tan(30 degrees) = X tan(60 degrees) + 1000 tan(60 degrees)
X tan(30 degrees) - X tan(60 degrees) = 1000 tan(60 degrees)
X(tan(30 degrees) - tan(60 degrees)) = 1000 tan(60 degrees)
X((sqrt(3)/3) - sqrt(3)) = 1000(sqrt(3))
X(sqrt(3)/3 - sqrt(3)) = 1000(sqrt(3))
X((sqrt(3) - 3)/3) = 1000sqrt(3)
X = 1000sqrt(3) * 3 / (sqrt(3) - 3)
X = 3000
Therefore, the man has 3000 meters still to walk before he is directly under the balloon.