Hot air balloon

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

Correct result:

D =  55.5071 m

Solution:

$$\begin{gathered} h=600 \\ {u}_1=38+20\mathrm{/}60={\displaystyle \frac{115}{3}}\doteq 38.3333 \\ {u}_2=1+16\mathrm{/}60={\displaystyle \frac{19}{15}}\doteq 1.2667 \\ {u}_3=u_1+u_2=38.3333+1.2667={\displaystyle \frac{198}{5}}=39.6 \\ x=h\mathrm{/}\tan u_1^{\circ }=h\mathrm{/}\tan 38.3333333333^{\circ }=600\mathrm{/}\tan 38.3333333333^{\circ }=600\mathrm{/}0.790697=758.82371 \\ \tan (u_2+u_1)=(r+h)\mathrm{/}x \\ r=x\cdot \tan u_3^{\circ }-h=x\cdot \tan 39.6^{\circ }-h=758.8237\cdot \tan 39.6^{\circ }-600=758.8237\cdot 0.827272-600=27.75357 \\ D=2\cdot r=2\cdot 27.7536=55.5071\text{ m} \end{gathered}$$

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