The angle of elevation of the top of a chimney from the top of the tower is $60{}^\circ $ and the angle of depression of the foot of the chimney from the top of the tower is $30{}^\circ $ . If the height of the tower is 40m, find the height of the chimney. According to the pollution control norms, the minimum height of a smoke emitting chimney should be 100 meters. State if the height of the above mentioned chimney meets the pollution norms.
VerifiedHint: Assume that the height of the cloud above the lake level is ‘h’. Draw a rough diagram of the given conditions and then use the formula $\tan \theta =\dfrac{\text{perpendicular}}{\text{base}}$ in the different right angle triangles and substitute the given values to get the height.
Complete step-by-step answer:
Let us start with the question by drawing a representative diagram of the situation given in the question.
According to the above figure:
AE is the chimney and BD is the tower. From the figure we can also see that BD is equal to ME.
We have assumed the height of the chimney as ‘h’. Therefore, AE = h meters. Also, assume that the distance BM is ‘y’ meters.
Now, in right angle triangle ABM,
$\angle ABM={{60}^{\circ }}$
We know that, $\tan \theta =\dfrac{\text{perpendicular}}{\text{base}}$. Therefore,
$\tan {{60}^{\circ }}=\dfrac{AM}{BM}$
Since, BD = ME = 40 m, because they are opposite sides of the rectangle BDEM. Therefore,
AM = AE – EM = h – 40.
\[ \Rightarrow \tan {{60}^{\circ }}=\dfrac{AM}{BM} \]
\[ \Rightarrow \tan {{60}^{\circ }}=\dfrac{h-40}{y} \]
\[ \Rightarrow y=\dfrac{h-40}{\tan {{60}^{\circ }}}.......................(i) \]
Now, in right angle triangle BME,
\[\angle EBM={{30}^{\circ }}\]
We know that, $\tan \theta =\dfrac{\text{perpendicular}}{\text{base}}$. Therefore,
$\tan {{30}^{\circ }}=\dfrac{EM}{BM}$
\[ \Rightarrow \tan {{30}^{\circ }}=\dfrac{40}{y} \]
\[ \Rightarrow y=\dfrac{40}{\tan 30{}^\circ }.......................(ii) \]
From equations (i) and (ii), we get,
$\dfrac{h-40}{\tan {{60}^{\circ }}}=\dfrac{40}{\tan {{30}^{\circ }}}$
Substituting $\tan {{30}^{\circ }}=\dfrac{1}{\sqrt{3}}\text{ and tan45}{}^\circ \text{=}\sqrt{3}$, we get,
$\dfrac{h-40}{\sqrt{3}}=\dfrac{\sqrt{3}\times 40}{1}$
By cross multiplication, we get,
$h-40=120$
$\Rightarrow h=160$
Therefore, the height of the chimney is 160 meters. Also, the height is greater than 100 m, so meets the pollution norms.
Note: One may get confused in removing the unknown variables. Therefore, remember that we have to find the value of variable ‘h’ and we have to remove all other variables. We have used the tangent of the given angle because we have to find the height and we have a common base in two right angle triangles that can be easily cancelled.
