Answer
$3.6\ mi$
Work Step by Step
1. Given angle $X=42^\circ10'=42.17^\circ$ and angle $Y=23^\circ30'=23.5^\circ$, we can find angle $Z=180^\circ-42.17^\circ-23.5^\circ=114.33^\circ$
2. Use the Law of Sines, we have $\frac{sinX}{x}=\frac{sinZ}{z}\longrightarrow x=\frac{z\ sinX}{sinZ}=\frac{12.2\ sin42.17^\circ}{sin114.33^\circ}\approx8.99$
3. The height of the balloon can be found as $h=x\ sinY=8.99sin23.5^\circ\approx3.6\ mi$
