Elevation 80866

Find the height of the tower when the geodetic measured two angles of elevation α=34° 30'' and β=41°. The distance between places AB is 14 meters.

Correct answer:

h =  46 m

Step-by-step explanation:

$$\begin{gathered} a=14\text{m} \\ \alpha =41°=41 \\ \beta =34°30^{\prime }=34°+\frac{30}{60}°=34.5°=34.5 \\ \tan \alpha =h/x \\ \tan \beta =h/(x+a) \\ x=h/\tan \alpha =h\cdot t_1 \\ {t}_1=1/\tan \alpha =1/\tan 41°=1/0.869287=1.15037 \\ {t}_2=\tan \beta =\tan 34.5°=0.687281=0.68728 \\ {t}_2\cdot (h\cdot t_1+a)=h \\ {t}_2\cdot h\cdot t_1+a\cdot t_2=h \\ h=\frac{a\cdot t_2}{1-t_2\cdot t_1}=\frac{14\cdot 0.6873}{1-0.6873\cdot 1.1504}\doteq 45.9558\text{m} \\ \text{ Verifying Solution: } \\ x=h\cdot t_1=45.9558\cdot 1.1504\doteq 52.8661\text{m} \\ {\alpha }_2=\frac{180°}{\pi }\cdot \arctan (\frac{h}{x})=\frac{180°}{\pi }\cdot \arctan (\frac{45.9558}{52.8661})=41{}^{\circ } \\ {\beta }_2=\frac{180°}{\pi }\cdot \arctan (\frac{h}{x+a})=\frac{180°}{\pi }\cdot \arctan (\frac{45.9558}{52.8661+14})=\frac{69}{2}=34.5{}^{\circ } \end{gathered}$$

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