Answer:
Approximately [tex]22.2\; \rm m[/tex].
Step-by-step explanation:
By sine rule, the length of each side of a triangle is proportional to the sine value of the angle opposite to that side. For example, in this triangle [tex]\triangle ABC[/tex], angle [tex]\angle A[/tex] is opposite to side [tex]BC[/tex], while [tex]\angle C[/tex] is opposite to side [tex]AB[/tex]. By sine rule, [tex]\displaystyle \frac{BC}{\sin{\angle A}} = \frac{AB}{\sin \angle C}[/tex].
It is already given that [tex]BC = 22.4\; \rm m[/tex] and [tex]\angle A = 58^\circ[/tex]. The catch is that the value of [tex]\angle C[/tex] needs to be calculated from [tex]\angle A[/tex] and [tex]\angle B[/tex].
The sum of the three internal angles of a triangle is [tex]180^\circ[/tex]. In [tex]\triangle ABC[/tex], that means [tex]\angle A + \angle B + \angle C = 180^\circ[/tex]. Hence,
[tex]\begin{aligned}\angle C &= 180^\circ - \angle A - \angle B \\ &= 180^\circ - 58^\circ - 65^\circ \\ &= 57^\circ\end{aligned}[/tex].
Apply the sine rule:
[tex]\begin{aligned} & \frac{BC}{\sin{\angle A}} = \frac{AB}{\sin \angle C} \\ \implies & AB = \frac{BC}{\sin{\angle A}} \cdot \sin \angle C \end{aligned}[/tex].
[tex]\begin{aligned}AB &= \frac{BC}{\sin{\angle A}} \cdot \sin \angle C \\ &= \frac{22.4\; \rm m}{\sin 58^\circ} \times \sin 57^\circ \\ &\approx 22.2\; \rm m\end{aligned}[/tex].
