Question

Two cargo ships spot a signal fire on a small island. The captains know they are 140 feet

Answer 1

The distance from Ship B to the signal fire at point C is $405.4 \mathrm{ft}$

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,

$\begin{aligned}\angle A+\angle B+\angle C &=180^{\circ} \\82^{\circ}+78^{\circ}+\angle C &=180^{\circ} \\160^{\circ}+\angle C &=180^{\circ} \\\angle C &=20^{\circ}\end{aligned}$

Now, we shall solve the problem using the law of sines,

$\frac{\sin A}{a}=\frac{\sin C}{c}$

Substituting the values, we have,

$\frac{\sin 82^{\circ}}{a}=\frac{\sin 20^{\circ}}{140}$

$\frac{0.9903}{a}=\frac{0.3420}{140}$

Simplifying, we get,

$a=\frac{0.9903 \times 140}{0.3420}$

$a=405.4$

Thus, the distance from Ship B to the signal fire at point C is $405.4 \mathrm{ft}$