Answer
$3.68$ miles
Work Step by Step
Step 1. Draw a diagram as shown in the figure (not to scale) where $h$ is the height of the balloon, $BC=1mile$
Step 2. As $AD\parallel BE$, we have $\angle ABE=20^\circ$ and $\angle ACE=22^\circ$
Step 3. In the right triangle of $\Delta ABE$, we have $tan20^\circ=\frac{h}{BE}$ and $BE=h\cdot cot20^\circ$
Step 4. In the right triangle of $\Delta ACE$, we have $tan22^\circ=\frac{h}{CE}$ and $CE=h\cdot cot22^\circ$
Step 5. As $BC=1mile$, we have $BE-CE=BC$ and $h\cdot cot20^\circ-h\cdot cot22^\circ=1$
Step 6. Solve the above equation as $(cot20^\circ- cot22^\circ)h=1$ or $(2.747-2.475)h=1$ which gives $h\approx3.68 miles$
