Three pillars

On a straight road, three pillars are 6 m high at the same distance of 10 m. At what angle of view does Vlado see each pillar if it is 30 m from the first and his eyes are at 1.8 m high?

Correct answer:

A =  11.4032 °
B =  8.5707 °
C =  6.8633 °

Step-by-step explanation:

$$\begin{gathered} n=3 \\ h=6\text{ m } \\ a=10\text{ m } \\ {l}_1=30\text{ m } \\ o=1.8\text{ m } \\ \tan \alpha =o:l_1 \\ \alpha ={\displaystyle \frac{180\mathrm{°}}{\pi }}\cdot \arctan (o\mathrm{/}{l}_1)={\displaystyle \frac{180\mathrm{°}}{\pi }}\cdot \arctan (1.8\mathrm{/}30)\doteq 3.4336\text{ ° } \\ \tan \beta =(h-o):l_1 \\ \beta ={\displaystyle \frac{180\mathrm{°}}{\pi }}\cdot \arctan ((h-o)\mathrm{/}{l}_1)={\displaystyle \frac{180\mathrm{°}}{\pi }}\cdot \arctan ((6-1.8)\mathrm{/}30)\doteq 7.9696\text{ ° } \\ A=\alpha +\beta =3.4336+7.9696=11.4032\text{°}=11\mathrm{°}24^{\mathrm{\prime }}12\mathrm{"} \end{gathered}$$

$$\begin{gathered} \theta =\arctan (o\mathrm{/}(l_1+a))+\arctan ((h-o)\mathrm{/}(l_1+a))=\arctan (1.8\mathrm{/}(30+10))+\arctan ((6-1.8)\mathrm{/}(30+10))\doteq 0.1496\text{ rad } \\ B=\theta \to \text{ °}=\theta \cdot {\displaystyle \frac{180}{\pi }}\text{ °}=0.1496\cdot {\displaystyle \frac{180}{\pi }}\text{ °}=8.571\text{ °}=8.5707\text{°}=8\mathrm{°}34^{\mathrm{\prime }}14\mathrm{"} \end{gathered}$$

$$\begin{gathered} \mathrm{\Psi }=\arctan (o\mathrm{/}(l_1+2\cdot a))+\arctan ((h-o)\mathrm{/}(l_1+2\cdot a))=\arctan (1.8\mathrm{/}(30+2\cdot 10))+\arctan ((6-1.8)\mathrm{/}(30+2\cdot 10))\doteq 0.1198\text{ rad } \\ C=\mathrm{\Psi }\to \text{ °}=\mathrm{\Psi }\cdot {\displaystyle \frac{180}{\pi }}\text{ °}=0.1198\cdot {\displaystyle \frac{180}{\pi }}\text{ °}=6.863\text{ °}=6.8633\text{°}=6\mathrm{°}51^{\mathrm{\prime }}48\mathrm{"} \end{gathered}$$

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