ANSWER:
9.4 miles
STEP-BY-STEP EXPLANATION:
The distance from tower A to the fire can be calculated using the law of sines.
The law of sine is the following:
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]
In this case the corresponding values would be:
a = 10 miles
The angle A we do not know it, we must calculate it, knowing that the sum of all the angles within a triangle is equal to 180°, therefore:
[tex]\begin{gathered} A+B+C=180 \\ A=180-B-C \\ A=180-42-64 \\ A=74\degree \end{gathered}[/tex]
A = 74°
B = 64°
b = distance from tower A to the fire
Replacing and solving:
[tex]\begin{gathered} \frac{10}{\sin74}=\frac{b}{\sin 64} \\ b=10\cdot\frac{\sin 64}{\sin 74} \\ b=9.4\text{ miles} \end{gathered}[/tex]
The distance from tower A to the fire is equal to 9.4 miles.
