The distance from Ship B to the signal fire at point C is [tex]405.4 \mathrm{ft}[/tex]
Explanation:
It is given that two ships A and B are at a distance 140 ft from each other.
The angle formed by ship A is 82º and the angle formed by ship B is 78º.
To determine the distance from ship B to the signal fire at point C, we need to know the another angle.
Hence, we have,
[tex]\begin{aligned}\angle A+\angle B+\angle C &=180^{\circ} \\82^{\circ}+78^{\circ}+\angle C &=180^{\circ} \\160^{\circ}+\angle C &=180^{\circ} \\\angle C &=20^{\circ}\end{aligned}[/tex]
Now, we shall solve the problem using the law of sines,
[tex]\frac{\sin A}{a}=\frac{\sin C}{c}[/tex]
Substituting the values, we have,
[tex]\frac{\sin 82^{\circ}}{a}=\frac{\sin 20^{\circ}}{140}[/tex]
[tex]\frac{0.9903}{a}=\frac{0.3420}{140}[/tex]
Simplifying, we get,
[tex]a=\frac{0.9903 \times 140}{0.3420}[/tex]
[tex]a=405.4[/tex]
Thus, the distance from Ship B to the signal fire at point C is [tex]405.4 \mathrm{ft}[/tex]
