Answer
The height of the tree is 327.0 feet.
Work Step by Step
Let us consider A to be the point where the angle of elevation is $66{}^\circ $.
Also consider B to be the point where the angle of elevation is $50{}^\circ $.
Let C be the point at the top of the tree.
The angle made at point C by the other two sides is given by:
$\begin{align}
& C=180{}^\circ -A-B \\
& =180{}^\circ -66{}^\circ -50{}^\circ \\
& C=64{}^\circ
\end{align}$
Using the law of sines, we will find a:
$\begin{align}
& \frac{a}{\sin \,A}=\frac{c}{\sin \,C} \\
& \frac{a}{\sin \,66{}^\circ }=\frac{420}{\sin \,64{}^\circ } \\
& a=\frac{420\times \sin \,66{}^\circ }{\sin \,64{}^\circ } \\
& a=426.9
\end{align}$
The height of the tree denoted by h is given by
$\begin{align}
& h=a\,\sin \,B \\
& =426.9\,\sin \,50{}^\circ \\
& h=327.0
\end{align}$
Therefore, the height of the tree is 327.0 feet.
