Answer
The height of the plane is 7203.6 feet
Work Step by Step
Let $C$ be the plane's position. Then A, B, and C form a triangle.
The angle $B = 180^{\circ}-57^{\circ} = 123^{\circ}$
The angle $C = 180^{\circ}- 52^{\circ}-123^{\circ} = 5^{\circ}$
We can use the law of sines to find the length of the side $AC$:
$\frac{AC}{sin~B} = \frac{AB}{sin~C}$
$AC = \frac{(AB)~sin~B}{sin~C}$
$AC = \frac{(950~ft)~sin~123^{\circ}}{sin~5^{\circ}}$
$AC = 9141.5~ft$
We can draw a vertical line from the plane's position straight down to the ground. The length of this line $h$ is the height of the plane. Let $D$ be the point where this line meets the ground.
Then A, C, and D form a right triangle. We can find the height $h$:
$\frac{h}{AC} = sin~52^{\circ}$
$h = (AC)~sin~52^{\circ}$
$h = (9141.5~ft)~sin~52^{\circ}$
$h = 7203.6~ft$
The height of the plane is 7203.6 feet
