River

From the observatory 11 m high and 24 m from the river bank, river width appears in the visual angle φ = 13°. Calculate width of the river.

Result

y =  29.5 m

Solution:

h=11 m x=24 m A=13 rad=13 π180 =13 3.1415926180 =0.22689 B=arctan(x/h)=arctan(24/11)1.141 C=A+B=0.2269+1.1411.3679 rad c=180/π C=180/3.1416 1.367978.3764  tanC=(x+y)/h y=h tan(C)x=11 tan(1.3679)2429.476229.5 mh=11 \ \text{m} \ \\ x=24 \ \text{m} \ \\ A=13 ^\circ \rightarrow\ \text{rad}=13 ^\circ \cdot \ \dfrac{ \pi }{ 180 } \ =13 ^\circ \cdot \ \dfrac{ 3.1415926 }{ 180 } \ =0.22689 \ \\ B=\arctan (x/h)=\arctan (24/11) \doteq 1.141 \ \\ C=A+B=0.2269+1.141 \doteq 1.3679 \ \text{rad} \ \\ c=180/\pi \cdot \ C=180/3.1416 \cdot \ 1.3679 \doteq 78.3764 \ ^\circ \ \\ \tan C=(x+y)/h \ \\ y=h \cdot \ \tan(C)-x=11 \cdot \ \tan(1.3679)-24 \doteq 29.4762 \doteq 29.5 \ \text{m}



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