Answer
The height of the balloon is 2.7 miles above the ground.
Work Step by Step
We can find the angle $C$:
$A+B+C = 180^{\circ}$
$C = 180^{\circ}-A-B$
$C = 180^{\circ}-24^{\circ}50'-47^{\circ}20'$
$C = 107^{\circ}50'$
We can use the law of sines to find the length of side $BC$:
$\frac{BC}{sin~A} = \frac{AB}{sin~C}$
$BC = \frac{AB~sin~A}{sin~C}$
$BC = \frac{(8.4~mi)~sin~24^{\circ}50'}{sin~107^{\circ}50'}$
$BC = 3.7~mi$
We can find the height $h$ of the balloon:
$\frac{h}{BC} = sin~47^{\circ}20'$
$h = (BC)~sin~47^{\circ}20'$
$h = (3.7~mi)~sin~47^{\circ}20'$
$h = 2.7~mi$
The height of the balloon is 2.7 miles above the ground.
