Hot air balloon

The center of the balloon is at an altitude of 600 m above the ground (AGL). The observer on earth sees the center of the balloon at an elevation angle of 38°20'. The balloon is seen from the perspective of an angle of 1°16'. Calculate the diameter of the balloon.

Correct answer:

D =  55.5071 m

Step-by-step explanation:

$$\begin{gathered} h=600 \\ {u}_1=38+20/60=\frac{115}{3}\doteq 38.3333 \\ {u}_2=1+16/60=\frac{19}{15}\doteq 1.2667 \\ {u}_3=u_1+u_2=38.3333+1.2667=\frac{198}{5}=39.6 \\ x=h/\tan u_1=h/\tan 38.333333333333°=600/\tan 38.333333333333°=600/0.790697=758.82371 \\ \tan (u_2+u_1)=(r+h)/x \\ r=x\cdot \tan u_3-h=x\cdot \tan 39.6°-h=758.8237\cdot \tan 39.6°-600=758.8237\cdot 0.827272-600=27.75357 \\ D=2\cdot r=2\cdot 27.7536=55.5071\text{m} \end{gathered}$$

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