Answer
$AB=118m$
Work Step by Step
$$B=112^\circ10',C=15^\circ20',BC=a=354m$$
$$AB=c=?$$
To find $c$, we look at the given information. $a$ is already known, so we can apply the following law of sines:
$$\frac{c}{\sin C}=\frac{a}{\sin A}$$
$$c=\frac{a\sin C}{\sin A}$$
In this equation, $a$ and $C$ are known, but $A$ is unknown.
However, we already know $B$ and $C$, so it is easy to calculate $A$ right away from the fact that the sum of 3 angles in any triangle equals $180^\circ$.
1) Find $A$
As the sum of 3 angles in any triangle equals $180^\circ$:
$$A+B+C=180^\circ$$
$$A+112^\circ10'+15^\circ20'=180^\circ$$
$$A+127^\circ30'=180^\circ$$
$$A=52^\circ30'=52.5^\circ$$
For calculation, we also need to change angle $C$ to complete degree:
$$C=15^\circ20'\approx15.333^\circ$$
2) Find $c$
$$c=\frac{a\sin C}{\sin A}$$
$$c=\frac{354\sin15.333^\circ}{\sin52.5^\circ}$$
$$c\approx118m$$
That means $AB=118m$
