Building elevation angle

In the building, I focused at an angle 30°. When I moved the 5 m building, I focused at an angle 45°. What is the height of the building?

Correct answer:

x =  6.83 m

Step-by-step explanation:

$$\begin{gathered} \alpha =30{}^{\circ } \\ \beta =45{}^{\circ } \\ \tan \alpha =x/y \\ \tan \beta =x/(y-5) \\ \tan \beta =x/(x/\tan \alpha -5) \\ \tan \beta (x/\tan \alpha -5)=x \\ \tan \beta (x-5\tan \alpha )=x\tan \alpha \\ x\tan \beta -5\tan \alpha \cdot \tan \beta =x\tan \alpha \\ x(\tan \beta -\tan \alpha )=5\tan \alpha \cdot \tan \beta \\ x=5\cdot \frac{\tan \alpha \cdot \tan \beta }{\tan \beta -\tan \alpha }=5\cdot \frac{\tan 30°\cdot \tan 45°}{\tan 45°-\tan 30°}=5\cdot \frac{0.57735\cdot 1}{1-0.57735}=6.83=6.83\text{m} \end{gathered}$$

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