A chimney $20$ metres high, standing on the top of a building subtends an angle whose tangent is $\frac{1}{6}$ at a distance $70$ metres from the foot of the building. Find the height of the building.

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Solution

As shwon in the fig. let $C B$ and $A B$ be the heights of the chimney and building respectively.
According to the given condition,
$t a n θ = \frac{1}{5}$ ,

$t a n β = \frac{H}{70}, t a n α = \frac{H + 20}{70}$

$α = θ + β$

$∴ t a n α = \frac{t a n θ + t a n β}{1 - t a n θ t a n β}$

or $\frac{H + 20}{70} = \frac{70 + 6 H}{420 - H}$

or $(20 + H) (420 - H) = 70 (70 + 6 H)$

or $8400 + 400 H - H^{2} = 4900 + 420 H$

or $H^{2} + 20 H - 3500 = 0$

or $(H + 70) (H - 50) = 0$

$\Rightarrow H = 50$

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