Answer
The height of the tree is $\text{233}\text{.0 feet}$.
Work Step by Step
By observing the figure, we can find
$\begin{align}
& C=62{}^\circ +23{}^\circ \\
& =85{}^\circ
\end{align}$
To find angle B, we will use the angle sum property:
$\begin{align}
& B=180{}^\circ -A-C \\
& =180{}^\circ -75{}^\circ -85{}^\circ \\
& =20{}^\circ
\end{align}$
Using the law of sines, we will find c:
$\begin{align}
& \frac{c}{\sin C}=\frac{b}{\sin B} \\
& \frac{c}{\sin 85{}^\circ }=\frac{80}{\sin 20{}^\circ } \\
& c=\frac{80\sin 85{}^\circ }{\sin 20{}^\circ } \\
& \approx 233.0
\end{align}$
Therefore, the height of the tree is $\text{233}\text{.0 feet}$.
