From the top of a building 'h' metre high, the angle of elevation of the top of a tower is found to be θ. from the bottom of the same building the angle of elevation of the top of the tower is found to be ø. show that the height of the tower is htanø/tanø-tanθ
Answers
Let the height of the tower be x
Express the distance between the two buildings from the top of the building:
tan θ = opp/adj
tan θ = (x - h)/distance
distance = (x - h)/tan θ
Express the distance between the two buildings from the bottom of the building:
tan ø = opp/adj
tan ø = x/distance
distance = x/tan ø
Since both the distance is the same:
(x - h)/tan θ = x/tan ø
Cross multiply:
(x - h) tan ø = x tan θ
expand (x - h):
xtan ø - htan ø = x tan θ
Add (htan ø) from both sides:
xtan ø = x tan θ + htan ø
Subtract (x tan θ) from both sides:
xtan ø - x tan θ = htan ø
Take out common factor x:
x(tan ø - tan θ) = htan ø
Divide both sides by (tan ø - tan θ):
x = htan ø / (tan ø - tan θ)
⇒ height of the tower = htan ø / (tan ø - tan θ) [ shown]
In the attachments I have answered this problem.
The solution is simple and easy to understand.
See the attachment for detailed solution.
