the angle of elevation to the top of temple from the point in the same horizontal plane is 30°,where observed from another which 150 meter straightly ahead from the original point on the same plane,the angle of the elevation of the top of the temple is 60°.find the height of the temple

Dear Student,

Please find below the solution to the asked query:

We form our diagram from given information , As :

Here , Height of Temple =  BC  =  h m  and AD  =  150 m and BD  =  x m 

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

In triangle ABC we get :

$$\begin{gathered} \tan \left(\angle \mathrm{BAC}\right)= \frac{\mathrm{BC}}{\mathrm{AB}} \\ \Rightarrow \tan 30°= \frac{h}{\mathrm{AD} + \mathrm{BD}} \\ \Rightarrow \frac{1}{\sqrt{3}}= \frac{h}{150 + x} ( \mathrm{We} \mathrm{know} : \tan 30° = \frac{1}{\sqrt{3}} ) \\ \Rightarrow 150 + x= \sqrt{3} h \\ \mathbf{\Rightarrow }\mathit{x}\mathbf{=}\mathbf{ }\sqrt{\mathbf{3}}\mathbf{ }\mathit{h}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{150}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{1}\mathbf{\hspace{0.17em}}\mathbf{)} \end{gathered}$$

And in triangle DBC we get :

$$\begin{gathered} \tan \left(\angle BDC\right)= \frac{\mathrm{BC}}{BD} \\ \Rightarrow \tan 60°= \frac{h}{x} \\ \Rightarrow \sqrt{3}= \frac{h}{x} ( \mathrm{We} \mathrm{know} : \tan 60° = \sqrt{3} ) \\ \Rightarrow x= \frac{h}{\sqrt{3}} , \mathrm{Substitute} \mathrm{that} \mathrm{in} \mathrm{equation} 1 \mathrm{we} \mathrm{get} : \\ \frac{h}{\sqrt{3}} = \sqrt{3} h - 150 \\ \Rightarrow h = 3 h - 150\sqrt{3} \\ \Rightarrow 2 h =150\sqrt{3} \\ \mathbf{\Rightarrow }\mathit{h}\mathbf{ }\mathbf{=}\mathbf{75}\sqrt{\mathbf{3}}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{75}\mathbf{\times }\mathbf{1}\mathbf{.}\mathbf{732}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{129}\mathbf{.}\mathbf{9}\mathbf{ }\mathbf{m} \end{gathered}$$
Therefore,

Height of Temple  =  75$\sqrt{3}$ = 129.9 m                                                      ( Ans )

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