Answer
The height of the leaning tree is 13.7 meters
Work Step by Step
Let A be the point at the bottom of the tree. Let B be the point on the ground where the angle of elevation is $66^{\circ}$. Let C be the point at the top of the tree. The points ABC form a triangle.
We can find the angle at point A:
$A = 90^{\circ}-8.0^{\circ} = 82^{\circ}$
The angle at point B is $66^{\circ}$
We can find the angle at point C:
$C = 180^{\circ}-82^{\circ} -66^{\circ} = 32^{\circ}$
We can use the law of sines to find the length between the bottom and top of the tree:
$\frac{AC}{sin~B} = \frac{AB}{sin~C}$
$AC = \frac{AB~sin~B}{sin~C}$
$AC = \frac{(8.0~m)~sin~66^{\circ}}{sin~32^{\circ}}$
$AC = 13.8~m$
We can find the height $h$ of the leaning tree:
$\frac{h}{AC} = cos(8.0^{\circ})$
$h = (AC)~cos(8.0^{\circ})$
$h = (13.8~m)~cos(8.0^{\circ})$
$h = 13.7~m$
The height of the leaning tree is 13.7 meters
