Clouds

From two points A and B on the horizontal plane was observed forehead cloud above the two points under elevation angle 73°20' and 64°40'. Points A , B are separated by 2830 m. How high is the cloud?

Result

x =  3662.055 km

Solution:

$u_{1}=73+20/60=\dfrac{ 220 }{ 3 } \doteq 73.3333 \ \\ u_{2}=64+40/60=\dfrac{ 194 }{ 3 } \doteq 64.6667 \ \\ t_{1}=\tan u_{1} ^\circ =\tan 73.3333333333^\circ \ =3.340233=3.34023 \ \\ t_{2}=\tan u_{2} ^\circ =\tan 64.6666666667^\circ \ =2.112335=2.11233 \ \\ s=2830 \ \\ a+b=s \ \\ x=a \cdot \ t_{1}=b \cdot \ t_{2} \ \\ a \cdot \ t_{1}=(s-a) \cdot \ t_{2} \ \\ a=s \cdot \ t_{2}/(t_{1}+t_{2})=2830 \cdot \ 2.1123/(3.3402+2.1123) \doteq 1096.3473 \ \\ b=s-a=2830-1096.3473 \doteq 1733.6527 \ \\ x=a \cdot \ t_{1}=1096.3473 \cdot \ 3.3402 \doteq 3662.055 \doteq 3662.055 \ \text{km}$

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