Steeple

A church tower is seen from the road at an elevation angle of 52°. When we move back a further 29 metres, it is seen at an elevation angle of 21°. How tall is the tower?

Final Answer:

h =  15.9 m

Step-by-step explanation:

$$\begin{gathered} n=29\text{m} \\ \alpha =52{}^{\circ } \\ \beta =21{}^{\circ } \\ \tan \alpha =h/x \\ \tan \beta =h/(x+n) \\ x=h/\tan \alpha =h\cdot t_1 \\ {t}_1=1/\tan \alpha =1/\tan 52°=1/1.279942=0.78129 \\ {t}_2=\tan \beta =\tan 21°=0.383864=0.38386 \\ {t}_2\cdot (h\cdot t_1+n)=h \\ {t}_2\cdot h\cdot t_1+n\cdot t_2=h \\ h=\frac{n\cdot t_2}{1-t_2\cdot t_1}=\frac{29\cdot 0.3839}{1-0.3839\cdot 0.7813}\doteq 15.9008\text{m} \\ \text{ Verifying Solution: } \\ x=h\cdot t_1=15.9008\cdot 0.7813\doteq 12.4231\text{m} \\ {T}_1=x/h=12.4231/15.9008\doteq 0.7813 \\ {T}_2=\frac{h}{x+n}=\frac{15.9008}{12.4231+29}\doteq 0.3839 \end{gathered}$$

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