# Lookout station B is located 9 mi due east of station A. The bearing of a fire from A is S 19W and the bearing from B is S 39.833W. How do you determine the distance from the fire to B (to the nearest tenth of a mile)?

Dec 14, 2016

Distance to Fire from station B: $\textcolor{g r e e n}{29.9 \text{ miles}}$

#### Explanation:

Here is my understanding of the situation:

$\angle F A B = {19}^{\circ} + {90}^{\circ} = {109}^{\circ}$
$\angle A B F = {90}^{\circ} - {39.833}^{\circ}$
$\angle A F B = {180}^{\circ} - \left(\angle F A B + \angle A B F\right) = {20.837}^{\circ}$

By the Law of Sines:
$\frac{\left\mid F B \right\mid}{\sin \left(\angle F A B\right)} = \frac{\left\mid A B \right\mid}{\sin \left(\angle A F B\right)}$

$\left\mid F B \right\mid = \frac{9 \text{ miles}}{\sin \left({20.837}^{\circ}\right)} \times \sin \left({109}^{\circ}\right)$

and with the aid of a calculator:
$\left\mid F B \right\mid \approx 29.923 \text{ miles}$
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