Answer
The distance from station A to the fire is 1.0 mile.
Work Step by Step
Here, the fire is denoted at point C at the distance b from station A. We will apply the sine rule to calculate the distance but at first we will calculate the angle.
By using the figure, calculate the angle $\angle A,\angle B $ and $\angle C $,
$\begin{align}
& \angle A\,=\,90{}^\circ -50{}^\circ \\
& =40{}^\circ
\end{align}$
$\begin{align}
& \angle B=\,90{}^\circ -40{}^\circ \\
& =50{}^\circ
\end{align}$
And,
$\begin{align}
& \angle C=\,180-50{}^\circ -40{}^\circ \\
& =90{}^\circ
\end{align}$
The distance of side c is given, so we will apply the sine law,
$\frac{b}{\sin B}=\frac{c}{\sin C}$
Here,
$ c=1.6\,\,\text{miles},\,\angle B=50{}^\circ \,$ and $\angle C=90{}^\circ $.
Now by putting the value in the equation, we get:
$\begin{align}
& \frac{b}{\sin 50{}^\circ }=\frac{1.6}{\sin 90{}^\circ } \\
& \frac{b}{0.766}=\frac{1.6}{1} \\
& b\approx 1.0\,\,\text{mile}
\end{align}$
