The shadow 2

The shadow of a tower standing on level ground is found to be 40 m longer when the Sun's altitude is 30° than when it is 60°. Find the height of the tower.

Final Answer:

h =  34.641 m

Step-by-step explanation:

$$\begin{gathered} \alpha =30{}^{\circ } \\ \beta =60{}^{\circ } \\ s=40\text{m} \\ \tan \beta =h:x \\ \tan \alpha =h:(x+s) \\ {t}_1=\tan \alpha =\tan 30°=0.57735 \\ {t}_2=\tan \beta =\tan 60°=1.732051=1.73205 \\ {t}_2=h:x \\ {t}_1=h:(x+s) \\ {t}_2\cdot x=h \\ {t}_1(x+s)=h \\ {t}_2\cdot x=t_1(x+s) \\ 1.7320508075689\cdot x=0.57735026918963(x+40) \\ 1.7320508075689\cdot x=0.57735026918963\cdot (x+40) \\ 1.154701x=23.094011 \\ x=\frac{23.09401077}{1.15470054}=20 \\ x=20 \\ h=t_2\cdot x=1.7321\cdot 20=20\sqrt{3}=34.641\text{m} \end{gathered}$$

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