Answer
The distance from the base to the top of the tower is about $\text{184}\text{.3 feet}$.
Work Step by Step
Let the distance from the base to the top of the tower -- that is, the distance between A and C -- be b.
To find angle C we will use the angle sum property:
$\begin{align}
& C=180{}^\circ -A-B \\
& =180{}^\circ -84.7{}^\circ -50{}^\circ \\
& =45.3{}^\circ
\end{align}$
Using the law of sines we will find c:
$\begin{align}
& \frac{c}{\sin C}=\frac{b}{\sin B} \\
& \frac{171}{\sin 45.3{}^\circ }=\frac{b}{\sin 50{}^\circ } \\
& b=\frac{171\sin 50{}^\circ }{\sin 45.3{}^\circ } \\
& \approx 184.3
\end{align}$
The distance between A and C is about $\text{184}\text{.3 feet}$.
