First, let's focus in this portion:
Let's use sine identity:
[tex]\begin{gathered} \sin (\theta)=\frac{opposite}{\text{hypotenuse}} \\ \sin (30)\frac{w}{8} \\ \text{Solving for w:} \\ w=8\cdot\sin (30)=4 \end{gathered}[/tex]
Now, let's focus in this portion:
Let's use tangent identity:
[tex]\begin{gathered} \tan (\theta)=\frac{opposite}{\text{adjacent}} \\ \\ \tan (30)=\frac{y}{4} \\ \text{solving for y:} \\ y=4\cdot\tan (30)=\frac{4\sqrt[]{3}}{3}\approx2.309 \end{gathered}[/tex]
Let's focus on the first portion again:
AB would be:
AB= y + z
Let's find z using tangent identity:
[tex]\begin{gathered} \tan (30)=\frac{4}{z} \\ \text{Solving for z:} \\ z=\frac{4}{\tan (30)}=4\sqrt[]{3}\approx6.928 \end{gathered}[/tex]
Therefore:
[tex]AB=\frac{4\sqrt[]{3}}{3}+4\sqrt[]{3}=\frac{16\sqrt[]{3}}{3}\approx9.2376[/tex]
