Answer
The fire is 15.0 miles from station A.
Work Step by Step
Let the fire be at point C. Let points A, B, and C make a triangle.
Therefore,
$\begin{align}
& A=90{}^\circ -56{}^\circ \\
& =34{}^\circ
\end{align}$
$\begin{align}
& B=90{}^\circ -23{}^\circ \\
& =67{}^\circ
\end{align}$
and
$\begin{align}
& \\
& C=180{}^\circ -34{}^\circ -67{}^\circ \\
& =79{}^\circ
\end{align}$
Now, by using the law of sines, we will find b:
$\begin{align}
& \frac{b}{\sin \,B}=\frac{c}{\sin \,C} \\
& \frac{b}{\sin \,67{}^\circ }=\frac{16}{\sin \,79{}^\circ }
\end{align}$
This implies that
$\begin{align}
& b=\frac{16}{\sin \,79{}^\circ }\times \sin \,67{}^\circ \\
& b=15.0
\end{align}$
Therefore, the fire is 15.0 miles from station A.
