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=tan 60 degrees and H=(X+1000) tan 30 degrees and use these two equations to find X
Let the distance between point A and B be D. From the information given, we can form a right triangle with the height of the balloon (H), the distance from point A to the balloon (X + D) and the distance from point B to the balloon (X).
Using trigonometry, we have:
tan(30 degrees) = H / (X + D)
tan(60 degrees) = H / X
From the first equation, we can express H in terms of tan(30 degrees):
H = (X + D)tan(30 degrees)
From the second equation, we can substitute the expression for H that we found in the first equation:
tan(60 degrees) = (X + D)tan(30 degrees) / X
√3 = (X + D) / X
Therefore, X + D = √3X
D = (√3 - 1)X
Given that D = 1000 meters, we can solve for X:
1000 = (√3 - 1)X
X = 1000 / (√3 - 1)
Now, we can solve for H:
H = (X + 1000)tan(30 degrees)
H = (1000 / (√3 - 1) + 1000)tan(30 degrees)
H = ((√3 + 1)1000)tan(30 degrees)
H = √3 * 1000
Therefore, H = √3 and X = 1000 / (√3 - 1).