a house 15 metres high stands on one side of a park from a point on the roof of the house the angle of depression of the foot of the chimney standing on the opposite side of the park is 30 and from the base of the housthe angle of deviation of the top of the chimney is 60. Find the height of the chimney and the distance in between.

Dear Student,

Please find below the solution to the asked query:

We form diagram from given information , As :



Here ,

Height of house =  AB  =  15 m ,

Height of chimney =  CD

And

Distance between house and chimney =  AC

We know : $\tan \mathrm{\theta } = \frac{\mathrm{Opposite}}{\mathrm{Adjacent}}$

In $∆$ ABC

$$\begin{gathered} \tan 30° = \frac{\mathrm{AB}}{\mathrm{AC}} \\ \Rightarrow \frac{1}{\sqrt{3}} = \frac{15}{\mathrm{AC}} ( \mathrm{We} \mathrm{know} \tan 30° =\frac{1}{\sqrt{3}} ) \\ \mathbf{\Rightarrow }\mathbf{AC}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{15}\sqrt{\mathbf{3}}\mathbf{ }\mathbf{m} \end{gathered}$$


In $∆$ ACD

$$\begin{gathered} \tan 60° = \frac{\mathrm{CD}}{\mathrm{AC}} \\ \Rightarrow \sqrt{3} = \frac{\mathrm{CD}}{15\sqrt{3}} ( \mathrm{We} \mathrm{know} \tan 60° =\sqrt{3} ) \\ \mathbf{\Rightarrow }\mathbf{CD}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{45}\mathbf{ }\mathbf{m} \end{gathered}$$
So,
Height of chimney =  45 m

And

Distance between house and chimney =  15$\sqrt{3}$ m                                                ( Ans )

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